rQA^ 

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UC-NRLF 


*B    417   10fi 


LIBRARY 

OF  THE 

University  of  California. 


Gl  FT    OF 


C/ass 


SYMPOSIUM   ON   MATHEMATICS 
FOR   ENGINEERING   STUDENTS 

BEING    THE 

PROCEEDINGS   OF  THE  JOINT  SESSIONS 

OF  THE 

CHICAGO  SECTION  OF 
/THE  AMERICAN  MATHEMATICAL  SOCIETY 

*"  AND 

SECTION  A,   MATHEMATICS,   AND 
SECTION  D,  MECHANICAL  SCIENCE  AND  ENGINEERING 

OF  THE 

AMERICAN  ASSOCIATION  FOR  THE  ADVANCEMENT  OF  SCIENCE 

HELD   AT 

THE    UNIVERSITY  OF   CHICAGO 
DECEMBER   30   AND   31,   1907 


Report  of  the  Meeting Wm.  T.  MagbudeB,  Secretary,  Ohio  State  University 

Present  Condition  of  Mathematical  Instruction  for  Engineers  in  American  Colleges 

Edgab  J.  Townsend,  University  of  Illinois 
The  Teaching  of  Mathematics  to  Engineering  Students  in  Foreign  Countries 

Alexander  Ziwet.  University  of  Michigan 

The  Teaching  of  Mathematics  for  Engineers Chas.  F.  Scott,  Pittsburg,  Pa. 

The  Point  of  View  in  Teaching  Engineering  Mathematics R.  S.  Woodward,  Carnegie  Institution 

The  Teaching  of  Mathematics  to  Students  of  Engineering: 
From  the  Standpoint  of  the  Practising  Engineer 

Ralph  Modjeskt,  Chicago,  111.;  J.  A.  L.  Waddell,  Kansas  City,  Mo. 
From  the  Standpoint  of  the  Professor  of  Engineering 

Gabdneb  S.  Williams,  University  of  Michigan  ;  Arthur  N.  Talbot,  University  of  Illinois  ; 
George  F.  Swain,  Massachusetts  Institute  of  Technology. 
From  the  Standpoint  of  the  Professor  of  Mathematics  in  the  Engineering  College 

Chas.  S  Sltchteb.  University  of  Wisconsin  ;  Fbedebick  S.  Woods,  Massachusetts  Institute 
of  Technology  ;  Feed  W.  McNaib,  Michigan  College  of  Mines. 
General  Discussion:  Calvin  M.  Woodward,  Washington  University  ;  B.  F.  Groat,  University  of  Min- 
nesota ;  C.  S.  Howe,  Case  School  of  Applied  Science  ;  Clarence  A.  Waldo,  Purdue  University  ; 
0.  P».  Williams,  Kalamazoo  College  ;  J.  Burkitt  Webb,  Stevens  Institute  of  Technology  ; 
FT.  T.  Eddy,  University  of  Minnesota  ;  S.  M.  Barton,  University  of  the  South  ;  Arthur  E. 
Haynes,  University  of  Minnesota  ;  Arthur  S.  Hathaway,  Rose  Polytechnic  Institute  ;  Edward 
V.  Huntington,  Harvard  University  ;  Donald  F.  Campbell,  Armour  Institute  of  Technology. 

This  symposium  was  arranged  for  by  a  committee  of  the  Chicago  Section  of  the  American  Mathematical  Society 
appointed  at  its  meeting  of  December  28,  1906,  and  consisting  of  E.  B.  VanVleck,  University  of  Wisconsin,  Chair- 
man ;  H.  E.  Rlaught,  University  of  Chicago,  Secretary  ;  E.  J.  Townsend,  University  of  Illinois;  Alexander  Ziwet, 
University  of  Michigan;  E.  B.  Skinner,  University  of  Wisconsin  ;  A.  G.  Hall,  Miami  University;  H.  L.  Rietz, 
University  of  Illinois  ;  together  with  the  co-operation  of  Wm.  T.  Magruder,  Ohio  State  University,  Secretary  of  Sec- 
lion  D,  Mechanical  Science  and  Engineering,  of  the  American  Association  for  the  Advancement  of  Science. 

Reprinted  from  Science,  x.  s.  Vol.  XXVIII.,  Nos.  707,  708,  709,  710,  713,  714,  July  17,  24,  31, 

August  7,  28,  and  September  4,  1908 


OF 


L 


[Reprinted  from  Science,  N.  S.,  Vol.  XXVIIL,  No.  707,  Pages  65,  67-79,  July  17,  No.  708, 

Pages  109-118,  July  24,  No.  709,  Pages  129-188,  July  81,  No.  710,  Pages  161-170,  August 

7,  No.  718,  Pages  257-268,  August  28,  No.  7U,  Pages  289-299,  September  4,  1908.'] 


THE  AMERICAN  ASSOCIATION  FOR  THE  ADVANCEMENT 

OF  SCIENCE 

SECTION  D— MECHANICAL  SCIENCE  AND  ENGINEERING 

ENGINEERING-MATHEMATICS  SYMPOSIUM 


The  meeting  of  the  section  for  organiza- 
tion was  held  in  Cobb  Hall  of  the  Uni- 
versity of  Chicago  on  December  30  and  31, 
1907,  and  January  1,  1908.  The  vice- 
president  of  the  section,  Olin  H.  Landreth, 
professor  of  civil  engineering,  Union  Uni- 
versity, acted  as  chairman  of  the  section. 

At  the  meeting  of  the  general  committee 
on  January  2, 1908,  on  the  recommendation 
of  the  sectional  committee,  Dr.  George  F. 
Swain,  professor  of  civil  engineering, 
Massachusetts  Institute  of  Technology, 
was  elected  vice-president  and  chairman  of 
the  section  for  the  ensuing  year;  and  Mr. 
George  W.  Bissell,  dean  of  engineering  and 
professor  of  mechanical  engineering,  Mich- 
igan Agricultural  College,  was  elected  sec- 
retary for  the  next  five  years. 

The  promotion  of  acquaintance  and 
personal  knowledge  was  an  important 
factor  in  the  success  of  the  meeting,  which 
was  in  large  part  due  to  the  labors  and 
foresight  of  Professor  H.  E.  Slaught,  of 
the  department  of  mathematics  of  the  Uni- 
versity of  Chicago,  and  Secretary  of  the 
Chicago  Section  of  the  American  Mathe- 
matical Society. 

A  subscription  dinner  for  engineers  and 
mathematicians  and  their  friends  brought 
about  one  hundred  persons  together  at 
Hotel  Del  Prado  on  Monday  evening,  De- 
cember 30.  The  speakers  at  the  dinner 
were  introduced  by  E.  B.  Van  Vleck,  pro- 
fessor of  mathematics,  University  of  Wis- 
consin, Chairman  of  the  Chicago  Section  of 


the  American  Mathematical  Society.  They 
were  Calvin  M.  Woodward,  dean  of  the 
School  of  Engineering  and  Architecture, 
Washington  University,  St.  Louis,  Mo.; 
Charles-  F.  Scott,  consulting  engineer  of 
the  Westinghouse  Electric  &  Manufactur- 
ing Co.,  Pittsburg,  Pa.;  George  F.  Swain, 
professor  of  civil  engineering,  Massachu- 
setts Institute  of  Technology,  Boston, 
Mass. ;  and  Edward  V.  Huntington,  assist- 
ant professor  of  mathematics,  Harvard 
University,  Cambridge,  Mass. 

The    first    session    of    the    engineering- 
mathematics  symposium  was  held  on  Mon- 
day   afternoon,    December   30.     Professor 
Van    Vleck    acted    as    chairman.     Four 
papers  were  presented,  as  follows: 
The  Present   Condition   of  Mathematical 
Instruction  for  Engineers  in  American 
Colleges:  Edgar  J.  Townsend,  professor 
of  mathematics,  University  of  Illinois. 

The  Teaching  of  Mathematics  to  Engineer- 
ing Students  in  Foreign  Countries: 
Alexander  Ziwet,  professor  of  mathe- 
matics, University  of  Michigan. 

The  Teaching  of  Mathematics  for  Engi- 
neers: Charles  F.  Scott,  consulting 
engineer,  Westinghouse  Electric  and 
Manufacturing  Co. 

The  Point  of  View  in  Teaching  Engineer- 
ing-Mathematics: Robert  S.  Woodward, 
president  of  the  Carnegie  Institution  of 
Washington. 
The  two  sessions,  held  on  the  morning 


180924 


SCIENCE 


and  afternoon  of  December  31,  were  de- 
voted to  a  symposium  on  the  question: 
"What  is  needed  in  the  Teaching  of 
Mathematics  to  Students  of  Engineering? 
(a)  Range  of  Subjects;  (&)  Extent  in  the 
Various  Subjects;  (c)  Methods  of  Prepara- 
tion; (d)  Chief  Aims."  The  speakers 
represented  three  phases  of  the  subject, 
namely:  (a)  Prom  the  standpoint  of  the 
practising  engineer;  (&)  from  the  stand- 
point of  the  professor  of  engineering;  (c) 
from  the  standpoint  of  the  professor  of 
mathematics  in  the  engineering  college. 

Professor  Landreth  and  Professor 
Slaught  were  the  chairmen  of  the  two 
sessions.  The  speakers  were  as  follows: 
Ralph  Modjeski,  consulting  engineer,  Chi- 
cago, 111.;  J.  A.  L.  Waddell,  consulting 
bridge  engineer,  Kansas  City,  Mo.;  Gard- 
ner S.  Williams,  professor  of  civil,  hy- 
draulic, and  sanitary  engineering,  Univer- 
sity of  Michigan;  Arthur  N.  Talbot,  pro- 
fessor of  municipal  and  sanitary  engineer- 
ing, University  of  Illinois;  George  F. 
Swain,  professor  of  civil  engineering, 
Massachusetts  Institute  of  Technology; 
Charles  S.  Slichter,  consulting  engineer, 
U.  S.  Reclamation  Service,  and  professor 
of  applied  mathematics,  University  of  Wis- 
consin; Frederick  S.  Woods,  professor  of 
mathematics,  Massachusetts  Institute  of 
Technology;  and  Fred  W.  McNair,  presi- 
dent of  the  Michigan  College  of  Mines. 

Following  the  presentation  of  the  four 
formal  papers,  and  of  the  eight  prepared 
discussions  above  recorded,  a  general  dis- 
cussion was  held  on  the  entire  subject. 
The  following  persons  took  part  in  this 
general  discussion:  Calvin  M.  Woodward, 
professor  of  mathematics  and  applied  me- 
chanics, Washington  University;  Ben- 
jamin F.  Groat,  professor  of  mechanics 
and  mathematics,  School  of  Mines,  Uni- 
versity of  Minnesota;  Charles  S.  Howe, 
president,  Case  School  of  Applied  Science; 
Clarence  A.  Waldo,  professor  of  mathe- 


matics, Purdue  University;  Clarke  B. 
Williams,  professor  of  mathematics,  Kala^ 
mazoo  College;  J.  Burkitt  Webb,  late 
professor  of  mathematics  and  mechanics, 
Stevens  Institute;  Henry  T.  Eddy,  pro- 
fessor of  mathematics  and  mechanics,  Col- 
lege of  Engineering,  University  of  Min- 
nesota; Arthur  E.  Haynes,  professor  of 
engineering-mathematics,  University  of 
Minnesota;  Arthur  S.  Hathaway,  professor 
of  mathematics,  Rose  Polytechnic  Institute ; 
Edward  V.  Huntington,  assistant  professor 
of  mathematics,  Harvard  University;  and 
Donald  F.  Campbell,  professor  of  mathe- 
matics, Armour  Institute  of  Technology. 

On  motion  of  Professor  Campbell,  the 
chairman  was  authorized  to  appoint  a  com- 
mittee of  three  persons,  they  to  increase 
their  number  to  fifteen,  to  be  chosen  from 
among  teachers  of  mathematics  and  engi- 
neering and  from  the  practising  engineers 
of  the  country;  and  this  committee  of 
fifteen  was  authorized  by  the  meeting  to 
take  into  consideration  the  whole  subject 
of  the  mathematical  curriculum  in  the 
engineering  and  technical  departments  of 
colleges  and  universities,  and  to  report  to 
the  Chicago  Section  of  the  American 
Mathematical  Society.  On  motion  of  Wm. 
T.  Magruder,  ex-secretary  of  the  Society 
for  the  Promotion  of  Engineering  Educa- 
tion and  professor  of  mechanical  engineer- 
ing, Ohio  State  University,  the  motion  was 
amended  that  the  committee  of  fifteen  shall 
submit  its  report  to  the  Society  for  the 
Promotion  of  Engineering  Education  at 
its  annual  meeting  in  the  summer  of  1909. 
The  motion  as  amended  was  unanimously 
adopted  by  those  present.  It  is  hoped 
that  at  the  meeting  of  the  society  in  1909, 
a  second  engineering-mathematics  sym- 
posium may  be  held. 

The  selection  of  this  important  com- 
mittee was  entrusted  to  Professor  Edward 
V.  Huntington,  Harvard  University,  Pro- 
fessor   Gardner    S.    Williams,    University 


SCIENCE 


of  Michigan,  and  Professor  Edgar  J. 
Townsend,  University  of  Illinois.  They 
will  select  the  remaining  members  of  the 
committee,  choose  a  chairman  and  secre- 
tary, and  determine  the  scope  of  the  in- 
vestigation that  they  will  make. 

The  papers  will  be  printed  in  Science  in 
the  next  few  weeks.  They  will  prove  to  be 
interesting  reading  to  those  engaged  in 
either  mathematical  or  engineering  work 
and  will  show  the  tendencies  of  the  thought 
of  the  meeting.  The  key-note  of  all  the 
discussions  was  that  we  need  more  sym- 
pathy and  knowledge  of  the  ideals,  aims 
and  work  of  the  other  fellow. 

The  meeting  was  without  doubt  the  best 
attended  that  the  sections  have  held  for 
many  years,  the  interest  never  seemed  to 
flag  and,  while  no  wonderful  contributions 
were  made  to  scientific  knowledge,  every 
one  went  away  feeling  either  that  he  had 
gained  much  information  as  to  the  other 
man's  point  of  view  concerning  scien- 
tifically instructing  engineering  students  in 
mathematics  and  of  the  wishes  and  needs 
of  the  engineering  instructor,  or  that  he 
appreciated  more  the  quality  of  work  that 
was  now  being  done  by  teachers  of  mathe- 
matics in  engineering  colleges. 

¥m.  T.  Magruder, 
Secretary,  Section  D 


PRESENT    CONDITION    OF   MATHEMATICAL 

INSTRUCTION   FOR   ENGINEERS   IN 

AMERICAN    COLLEGES1 

Our  country  has  witnessed  in  recent 
years  a  most  marvelous  industrial  expan- 
sion and  development.  Along  with  this 
movement  has  come  a  rapidly  increasing 
demand  for  trained  men,  equipped  with  all 
that  science  can  contribute,  to  direct  and 

1  Opening  address  before  the  joint  meeting  of 
Sections  A  and  D  of  the  American  Association 
for  the  Advancement  of  Science  with  the  Chicago 
Section  of  the  American  Mathematical  Society 
for  the  discussion  of  the  topic  "  Mathematical 
Training  for  Engineers." 


carry  forward  this  development  of  our 
natural  resources  and  our  industrial  power. 
In  meeting  this  demand  our  technical 
schools  have  experienced  a  remarkable 
growth,  and  not  a  little  of  the  educational 
thought  and  activity  of  the  country  is 
being  directed  toward  the  problems  con- 
nected with  technical  instruction.  Well- 
equipped  engineering  schools  have  grown 
up  in  the  larger  centers  of  population  and 
most  of  the  larger  state  universities  now 
include  strong  engineering  departments. 
Mathematics  is  so  fundamental  to  all  of 
this  work,  and  so  large  a  proportion  of  the 
students  now  receiving  mathematical  in- 
struction in  this  country  anticipate  making 
use  of  it  later  in  connection  with  engineer- 
ing work,  that  it  has  been  thought  best  by 
the  Chicago  Section  of  the  American 
Mathematical  Society  to  invite  to  a  joint 
discussion  of  the  "Mathematical  Training 
of  Engineering  Students,"  representatives 
from  some  of  the  leading  engineering 
schools  and  some  of  those  consulting  engi- 
neers whose  wide  experience  has  brought 
them  into  contact  with  demands  of  actual 
practise. 

That  we  may  all  know  what  the  actual 
conditions  are  with  respect  to  this  instruc- 
tion and  consequently  have  some  common 
basis  for  our  discussion  and  our  conclu- 
sions, I  have  been  asked  to  present  a  state- 
ment of  the  work  in  mathematics  which 
is  now  being  given  to  engineering  students. 

As  the  basis  of  our  consideration,  I  have 
selected  seventeen  institutions  where  engi- 
neering work  is  an  important  feature.  Of 
these,  eight  give  their  attention  largely  or 
exclusively  to  technical  work,  and  the  re- 
maining institutions  have  strong  engineer- 
ing departments ;  so  that  the  mathematical 
work  given  in  these  institutions  may  be 
said  to  fairly  represent  the  preparation  in 
this  subject  for  engineering  students  in 
American  institutions. 

The  three  most  important  factors  enter- 


SCIENCE 


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p  =  through  progressions,  q  =  through  quadratics,  L  =  logarithms,  e  =  elective. 


ing  into  the  consideration  of  our  topic  are : 
the  entrance  requirements,  the  require- 
ments for  graduation,  and  the  qualifica- 
tions of  the  instructional  force. 

As  will  be  seen  from  Table  I.,  all  of 
these  seventeen  institutions  require  for 
entrance  algebra  through  quadratics,  to- 
gether with  plane  and  solid  geometry. 
Five  of  the  institutions  require  plane  trig- 
onometry, while  at  several  others  it  may  be 
counted  for  entrance  if  the  student  so 
elects.  It  will  be  observed  that  four  insti- 
tutions require  elementary  algebra  through 
progressions,  four  require  the  subject  of 
logarithms,  and  two,  Sheffield  and  Cornell, 
require  the  whole  of  college  algebra. 

There  is  a  general  tendency  over  the 
country  to  increase  rather  than  diminish 
the  entrance  requirements  in  mathematics. 
Several  institutions  have  recently  done  so, 
and  at  a  number  of  others  there  is  a  feel- 
ing that  both  trigonometry  and  college 
algebra  should  be  required.  This  disposi- 
tion to  increase  the  entrance  requirements 
has  come  about  not  so  much  because  of  a 
feeling  that  these  subjects  can  be  as  well 
or  better  taught  in  the  secondary  schools, 
but  because  of  a  feeling  on  the  part  of 
the  technical  schools  that  the  entrance  re- 
quirements should  be  made  as  high  as 
possible  in  order  to  give  room  in  the  cur- 


riculum for  those  professional  and  tech- 
nical branches  which  are  now  deemed 
essential.  It  may  well  be  questioned 
whether  we  are  not  in  some  danger  of 
going  too  far  in  increasing  the  require- 
ments. I  am  sure  that  we  should  all  agree 
that  the  guiding  principles  should  be  the 
limitations  of  the  secondary  school  pro- 
gram and  the  ability  of  the  pupil  at  that 
stage  of  his  maturity  to  readily  grasp  in 
a  comprehensive  manner  the  subjects  pre- 
sented. For  example,  the  advisability  of 
adding  college  algebra  to  the  entrance  re- 
quirements is  certainly  open  to  the  ob- 
jection that  portions  of  it  are  clearly  be- 
yond the  maturity  of  the  average  high 
school  pupil,  and  the  introduction  of  plane 
trigonometry  would  seem  inadvisable  in  the 
average  high  school  on  the  accredited  list 
of  the  state  universities  of  the  Mississippi 
Valley.  When  either  of  the  fundamental 
principles  mentioned  is  violated,  we  shall 
have  coming  to  our  freshman  class,  stu- 
dents with  a  decided  and  a  justifiable  dis- 
like for  anything  mathematical.  Bather 
than  to  encounter  this  danger,  it  would  be 
far  better  to  extend  the  engineering  course 
over  five  years  or  to  require  a  year  of 
college  work  in  science  and  mathematics 
before  the  student  enters  upon  his  tech- 
nical course.     In  this  connection,  it  is  in- 


SCIENCE 


teresting  to  note  that  the  University  of 
Minnesota  has  recently  extended  its  course 
to  five  years  for  students  in  civil,  me- 
chanical and  electrical  engineering,  dis- 
tributing the  required  work  in  mathe- 
matics throughout  the  first  four  years. 

The  writer  does  not  share  with  some  the 
feeling  that  a  greater  uniformity  in  en- 
trance requirements  is  either  desirable  or 
of  any  particular  consequence.  Each  in- 
stitution, and  especially  the  state  institu- 
tions, must  take  into  consideration  what 
the  secondary  schools  contributary  to  it 
can  do  satisfactorily  and  then  shape  its 
work  accordingly.  The  size  of  the  city, 
the  general  interest  in  educational  affairs, 
the  trend  which  local  interests  give  to  the 
public-school  curriculum,  all  tend  to  make 
it  possible  to  accomplish  in  one  community, 
or  in  one  section  of  the  country,  what 
would  be  quite  impossible  in  another.  We 
must  accept  our  students  with  such  prepa- 
ration as  our  normal  constituency  can  give, 
stimulated,  to  be  sure,  and  to  a  certain 
extent  guided  by  the  higher  institution  of 
learning,  and  build  our  technical  courses 
upon  that  preparation  as  best  we  may. 

More  general  dissatisfaction  is  expressed 
with  reference  to  the  preparation  of  our 
students  in  algebra  than  in  any  other  sub- 
ject. This  comes  from  both  eastern  and 
western  institutions  as  well  as  from  those 
of  the  Mississippi  Valley.  At  the  Uni- 
versity of  Illinois  last  year  forty  per  cent, 
of  the  freshman  class  failed  to  pass  a  quiz 
covering  the  main  points  of  elementary 
algebra  and  that  after  a  two  weeks'  re- 
view of  the  subject,  and  twenty-three  per 
cent,  of  the  class  failed  on  a  second  exami- 
nation some  weeks  later.  Of  the  one  hun- 
dred and  ninety  students  who  failed  on 
the  first  test,  seventy- four  per  cent,  entered 
the  university  without  conditions  from 
schools  where  the  work  had  been  examined 
and  approved  by  the  high-school  visitor. 
The  poor  results  which  we  get  in  algebra 


are  not  due,  in  my  estimation,  exclusively 
to  poor  instruction  in  the  subject  or  to  the 
lack  of  attention  in  the  high  school.  It 
is  the  one  subject  in  mathematics  which  is 
begun  in  the  high  school  and  completed 
in  the  college  course.  Often  the  high- 
school  algebra  is  completed  in  the  sopho- 
more year  and  then  not  taken  up  again 
until  the  student  enters  upon  his  technical 
course.  All  know  how  difficult  it  is  to 
retain  the  details  of  any  course  of  study 
during  an  interval  of  several  years 
in  which  the  subject  has  been  but  little 
used.  That  this  lapse  of  time  between  the 
completion  of  the  high-school  work  and 
the  beginning  of  the  college  work  is  an 
important  element  in  the  case  is  shown  by 
the  fact  that  of  the  one  hundred  and 
ninety  failures  mentioned  over  fifty  per 
cent,  had  not  had  algebra  for  at  least  four 
years,  and  only  ten  per  cent,  had  studied 
the  subject  the  year  before. 

A  substantial  gain  would  be  made  if  we 
should  urge  upon  the  high  schools  the 
desirability  of  putting  the  last  half  year 
devoted  to  algebra  in  the  senior  year  of 
the  high-school  curriculum  and  include 
in  that  work  the  more  difficult  parts  of 
the  subject  as  well  as  a  general  review 
of  the  parts  presented  earlier.  This 
arrangement  has  become  quite  common  in 
Illinois,  and  the  best  argument  that  can  be 
presented  in  favor  of  such  an  arrangement 
is  that  of  the  one  hundred  and  ninety  cases 
of  failure  cited  over  sixty-three  per  cent, 
had  completed  the  work  in  the  sophomore 
year  and  less  than  eight  per  cent,  had  had 
any  work  in  algebra  in  the  senior  year. 
Similar  records  have  been  kept  at  Illinois 
for  the  past  seven  or  eight  years  and  the 
data  given  are  typical  of  the  other  years. 

Unfortunately,  we  can  have  no  assur- 
ance that  when  a  student  has  once  mastered 
a  subject,  he  will  forever  afterwards  retain 
it.  Neither  can  we  hope  that  algebra  will 
ever  be  anything  other  than  the  weakest 


G 


SCIENCE 


place  in  the  preparation  of  our  students 
so  long  as  the  present  division  of  the  sub- 
ject so  largely  prevails.  It  is  a  situation 
which  we  must  accept,  and  the  only  thing 
we  can  do  is  to  make  such  recommenda- 
tions as  will  tend  to  reduce  the  number  of 
fatalities  as  the  boy  passes  from  his  sec- 
ondary school  to  his  technical  course.  The 
technical  school  must  expect  to  commence 
its  course  in  college  algebra  by  a  brief 
review  of  the  important  points  covered  in 
the  high  school,  by  taking  a  back-stitch,  so 
to  speak,  into  the  work  already  done. 
Most  of  the  western  schools  admit  by  cer- 
tificate to  the  freshman  class,  and  when  a 
pupil  is  once  graduated  from  an  accredited 
school,  he  has  earned  the  right  to  com- 
mence upon  his  technical  course.  At  the 
University  of  Illinois,  the  problem  has 
been  solved  by  saying  to  the  freshmen  in 
mathematics  that  while  there  is  no  disposi- 
tion to  deprive  them  of  their  entrance 
credit,  the  department  of  mathematics  may 
nevertheless  determine  the  conditions 
under  which  credit  in  college  algebra  can 
be  secured.  Accordingly,  those  students 
who  fail  to  pass  the  review  quiz  are  re- 
quired to  take  two  additional  hours  per 
week  in  the  subject  for  the  remainder 
of  the  semester  in  order  to  earn  the  same 
credit  that  is  given  to  others  at  the  close 
of  the  course.  This  has  the  advantage  of 
placing  all  of  the  students  practically  upon 
the  same  basis,  so  far  as  attainments  in 
algebra  are  concerned,  when  they  enter 
upon  the  second  semester's  work. 

A  somewhat  similar  plan  as  that  out- 
lined here  is  followed  also  at  the  Univer- 
sity of  Wisconsin,  and  perhaps  at  other 
institutions.  It  will  be  seen  from  Table 
I.  that  a  large  number  of  technical  schools 
are  now  requiring  work  in  logarithms  for 
entrance.  This  might  very  well  be  intro- 
duced in  connection  with  theory  of  ex- 
ponents and  used  with  advantage  in  high- 
school  physics.     It   is   also   gratifying  to 


observe  that  the  more  recent  texts  on 
algebra  provide  work  in  the  use  of  the 
graph  and  in  the  plotting  of  curves.  It  is 
very  desirable  that  the  work  in  elementary 
algebra,  including  the  work  of  curve-plot- 
ting, should  also  include  applications  to 
some  of  the  simpler  phenomena  studied  in 
the  high-school  course  in  physics,  and  this 
again  is  made  a  feature  in  some  of  the 
more  recent  texts.  Such  an  arrangement 
affords  an  additional  reason  for  putting 
some  of  the  work  in  algebra  late  in  the 
high-school  course  in  order  that  it  may 
follow  rather  than  precede  the  work  in 
physics,  thus  making  it  possible  to  intro- 
duce a  wider  range  of  physical  applica- 
tions than  could  otherwise  be  done. 

In  Table  II.  is  shown  the  number  of 
restrictions  given  in  each  of  the  various 
mathematical  subjects  required  of  engi- 
neering students.  The  average  number 
given  to  each  subject  for  the  seventeen 
institutions  is  approximately  as  follows: 
college  algebra  50,  plane  trigonometry  46, 
analytic  geometry  80,  and  calculus  130. 
In  a  number  of  the  institutions  named, 
spherical  trigonometry  is  taught  by  one  of 
the  engineering  departments,  usually  the 
civil-engineering  department,  in  connec- 
tion with  its  applications  to  geodesy.  The 
number  of  recitations  assigned  to  cal- 
culus usually  includes  also  a  short  course 
in  differential  equations.  In  two  cases 
where  a  course  of  more  than  usual  length 
in  the  subject  is  given  for  the  students  of 
a  particular  engineering  department,  the 
subject  has  been  listed  separately.2  One 
institution,  Rose  Polytechnic  Institute,  is 
unique  among  strictly  engineering  schools 
in  offering  throughout  the  four  years  of 
undergraduate  work  a  rather  large  amount 
of  elective  mathematics,  including  short 
courses  in  advanced  calculus,  least  squares, 

*  Table  III.  shows  the  number  of  recitations 
given  to  differential  equations  in  each  case  when 
that  subject  was  reported  separately. 


SCIENCE 


u 

a 
o 

a 

"3 
a 

o 

a 

to 

2 

* 

i 

03 
O 

8 

'u 

3 
o 

s 

E 

u 

71 
00 

o 

2 

"3 
S0 

3 

i 

a 
o 

■ 

1 

o 
O 

M 

Hi 

3 
* 

a 

3 

s 

• 

3 

e 

B 
M 

« 

.a 

GO 

s 

CO 

C 

o 

65 

45 

55 

/40 

\C.E. 

30 

36 

70 

36 

36 

90 

55 

72 

risi 

15 

H 

64 

Plane     Trigo- 

nometry  

60 

35 

30 

45 

45 

54 

70 

29 

54 

i  j 

45 

Spherical  Trig- 
onometry  

10 

/22 

\a 

80 
fo8 
1  C.E. 

no 

ICE. 

25 

10 

20 

11 

[18] 

10 

3 

Analytic  Ge- 

55 

100 

60 

90 

60 

108 

110 

90 

108 

72 

60 

54 
[18] 

90 

69 

90 

64 

155 

125 

120 

144 

108 
/96 
ICE. 

90 

144 

110 

180 

126 

144 

70 

180 

[72] 

100 

144 

180 

ri6o 

ICE. 

96 

Least  squares- 

/48 
\C.E. 

[18] 

Vector    Anal- 

/32 
ICE. 

ysis  

Projective  Ge- 

ometry  

|18| 

Dua tern  ions 

|18| 

differential 

J34 
)M.E. 

/45 
\E.E. 

Equations.... 

C  E.  =  civil  engineers,  M.  E.  =  mechanical  engineers,  E.  E.  =  electrical  engineers,   [  ]  =  elective. 

*  Massachusetts  Institute  of  Technology  now  offers  a  course  which  combines  the  instruction  in  algebra, 
analytic  geometry  and  calculus  rather  than  teaching  these  subjects  as  separate  branches.  In  tables 
II.  and  III.,  the  distribution  of  time  formerly  given  to  these  subjects  is  indicated  as  showing  better 
the  relative  emphasis  placed  upon  each. 


projective  geometry  and  quaternions.  In 
all  of  the  universities  listed,  and  at  the 
Massachusetts  Institute  of  Technology  the 
mathematical  department  offers  a  rather 
wide  range  of  advanced  subjects,  all  of 
which  are  open  to  engineering  students  so 
far  as  the  demands  of  their  technical 
course  will  permit. 

By  a  study  of  Table  II.,  it  will  be  seen 
that  a  considerable  difference  exists  in  the 
amount  of  attention  given  to  the  various 
subjects.  In  making  comparisons  in 
algebra  and  trigonometry,  however,  the 
difference  in  entrance  conditions  must  be 
taken  into  consideration.  The  amount  of 
work  given  in  algebra  ranges  from  fifteen 
recitations  at  Stevens  Institute,  with  an 
entrance  requirement  of  elementary  alge- 
bra through  progressions,  to  ninety  recita- 
tions at  Purdue  with  a  requirement  of  ele- 


mentary algebra  through  quadratics  for 
entrance.  Likewise  the  work  in  plane  tri- 
gonometry ranges  from  thirty  recitations 
at  the  Massachusetts  Institute  to  seventy  at 
Purdue.  Analytic  geometry  and  calculus 
are  naturally  the  most  important  subjects 
for  engineers  in  the  mathematical  curric- 
ulum. One  would  naturally  expect  to 
find  a  greater  uniformity  here.  This, 
however,  is  not  the  case.  In  analytic 
geometry,  it  will  be  noticed  that  Armour 
Institute  requires  but  fifty-five  recitations, 
while  the  University  of  Minnesota  gives 
one  hundred  and  ten  recitations  to  the  sub- 
ject. Again  in  calculus  the  work  varies 
from  seventy  recitations  at  Rensselaer  to  a 
maximum  of  one  hundred  and  eighty  at 
Missouri,  Wisconsin  and  Rose.  A  word 
should  be  said,  perhaps,  concerning  the 
number  of  recitations  recorded  in  the  case 


8 


SCIENCE 


of  Rensselaer.  The  department  of  mathe- 
matics of  that  institution  reports  that  the 
recitations  are  from  an  hour  and  a  quarter 
to  an  hour  and  a  half  in  length  and  that 
the  efficiency  of  the  work  is  still  further 
increased  by  the  fact  that  but  two  aca- 
demic studies  are  carried  simultaneously. 

It  will  be  of  interest  also  to  compare  the 
total  amount  of  time  spent  upon  mathe- 
matics at  these  various  institutions.  As 
will  be  seen  from  Table  III.,  this  ranges 
from  one  hundred  and  eighty  recitations 
at  Cornell  to  three  hundred  and  ninety-six 
at  Purdue.  In  making  this  comparison, 
we  should  again  take  into  consideration 
the  difference  in  entrance  requirements. 
When  this  is  done,  the  difference  is  more 
apparent  than  real.  For  example,  if  we 
add  to  the  number  of  recitations  given  at 
Cornell  the  number  of  recitations  given  at 
Purdue  to  college  algebra  and  trigonom- 
etry, which  are  required  for  entrance  at 
Cornell  as  compared  with  three  hundred 
and  ninety-six  at  Purdue. 

It  would  seem  that  the  technical  schools 
generally  might  well  afford  to  make  more 
ample  provision  for  elective  mathematics. 
Such  courses  as  spherical  trigonometry, 
least  squares,  differential  equations,  might 
well  be  placed  in  such  a  list.  In  this  way 
certain  subjects  which  are  desirable  for 
some  branches  of  engineering,  but  not  so 
essential  for  others,  could  be  taken  by 
those  students  interested.  Sheffield  offers 
as  an  elective  another  course  which  might 
be  given  with  advantage  at  other  technical 
schools,  namely,  a  course  in  scientific  com- 
putation in  which  the  use  of  modern  cal- 
culating machines  of  various  kinds  is 
explained  and  made  use  of.  It  would  also 
be  well  if  the  stronger  institutions  could 
go  still  farther  and  introduce  elective 
courses  in  spherical  harmonics,  vector  an- 
alysis, theory  of  functions  and  the  mathe- 
matical theory  of  heat,  electricity,  etc.,  to 
the  end  that  the  student  with  exceptional 


mathematical  ability  might  lay  a  broader 
foundation  for  the  theoretical  side  of  engi- 
neering. In  this  connection,  it  may  well 
be  questioned  whether  the  technical  schools 
of  this  country  are  in  general  offering  suf- 
ficient opportunity  for  that  training  which 
has  made  it  possible  for  such  men  as  Stein- 
metz,  Osborne  Reynolds  and  Stodola  to 
accomplish  the  work  which  has  made  them 
famous. 

Table  III.  shows  also  the  sequence  and 
the  distribution  by  years  of  the  required 
work  in  mathematics.  "We  are  quite  as 
much  interested,  however,  in  the  character 
as  in  the  amount  and  distribution  of  the 
mathematical  instruction  given  to  engineer- 
ing students.  The  close  observer  will  have 
noticed  the  change  which  has  been  made 
and  is  now  being  made  in  this  respect.  In 
recent  years  there  has  swept  over  the  coun- 
try a  wave  of  enthusiastic  discussion  con- 
cerning a  closer  and  better  correlation  of 
mathematics  with  the  physical  sciences. 
This  has  been  due  for  the  most  part  to  the 
influence  felt  in  this  country  of  the  Perry 
movement  in  England.  Much  is  to  be 
learned  from  this  movement,  and  still  more 
is  to  be  avoided.  The  discussions  which 
have  arisen  from  it  have  on  the  whole  had 
a  beneficial  effect  upon  the  teaching  of 
mathematics  both  in  America  and  in  Eng- 
land. 

It  has  first  of  all  led  to  the  introduction 
into  our  text-books,  and  still  more  gener- 
ally into  our  teaching,  of  a  very  much 
better  selection  of  problems— problems 
which  widen  the  student's  fund  of  infor- 
mation of  physical  phenomena  and  apply 
the  mathematical  principles  which  he  is 
acquiring  more  extensively  than  was  for- 
merly the  case  to  the  physical  laws  with 
which  he  is  familiar.  Such  problems  as 
the  following,  taken  from  a  recent  number 
of  an  educational  journal  purporting  to 
serve  the  interests  of  mathematical  teach- 
ers in  the  secondary  schools,  is  no  longer 


SCIENCE 

9 

TABLE 

m 

Institution 

Freshman 

Sophomore 

Junior 

Senior 

Total 

Al  65  ;  An  55  ;  C  50 

C  85  ;  Diff.  eqs.  20 

275 

Al  45  ;  Tr  70  ;  An  55 

An  45  ;  C  125 

Diff.  eqs.  34  (E.E.) 

Least  squares 
48  (C.E.) 

M.E.;  340 

E.E  •  374 

C.E.;  388 

An  60;  C  120 

Al  55  ;  Tr  35  ;  An  90 

C144 

180 
324 

An  80  ;  C  108 

Al  40  ;  Tr  22  ;  M  25 

(C.E.)  An  58;  C  96 

E.E.;  M.E.;  188 
C.E.;  241 

Mass.  Inst.. 

Al  30  ;  Tr  30  ;  An  60 

C  90; 
SphTrlO  (C.E.) 

Diff.  eqs.  45  (E.E.) 

M.E.;  210 
E.E.;  255 
C.E.;  220 

Michigan... 
Minnesota.. 
Missouri.... 

Al  36  ;  An  108 

Al  70  ;  Tr  70  ;  An  40 

Al  36  ;  Tr  55  ;  An  90 

C126 

An  70  ;  C  110 

C180 

Diff.  eqs.  18 

288 
360 
360 

Nebraska... 

A136;Tr54;  An 72; 
C18 

An  36 ;  C  108 

324 

Purdue 

Kensselaer.. 

Al 90 ;  Tr  90 

Al  45  ;  Tr  40  ;  An  39 

An  72 ;  C  72 
An  21  ;  C  70 

C72 

396 
225 

A172;  Tr54;  C  36 
[S  Tr  18]  ;  [Proj. 
Geom.  18]  [Al  18] 

An  54  ;  C  90 
[Quat  18]  [An  18] 

An  Dyn  54  (calculus) 

[C72] 

[Least  squares  18] 

360 

[180] 

Sheffield  .... 

An  90 

Logl5;SphTrl0; 
An  43  ;  C  30 

C100 

An  26  ;  C  86 

Diff.  eqs.  28 

[Probs.  and 
computing] 

190 
[Probs.] 

238 

"Wisconsin .. 

Al  and  Tr  90  ;  An  90 

C160; 

Diff.  eqs.  20  (M.E.; 
E.E.) 

M.E.;  E.E.;  360 
C.E.;  340 

Worcester... 

Al  64  ;  Tr  48 

An  64  ;  C  96 

Vect.  and  32  (E.E.) 

272 
E.E.;  304 

Al  =  algebra,  An  =  analytic  geometry,  C  =  calculus,  Tr  =  trigonometry,  M  =  mensuration, 
Quat  =  quaternions,  An  Dyn  =  analytical  dynamics,    [  ]  =  elective. 


thought  to  be  in  good  form  by  our  best 
instructors : 

"I  bought  674,867  sheep  at  less  than  $10 
per  head;  I  paid  for  them  in  ten-dollar 
bills  and  received  back  in  change  $7.39. 
How  many  bills  did  I  give  ? ' ' 

Need  I  call  attention  to  the  absurdity  of 
putting  such  problems  into  the  hands  of 
pupils?  How  many  farmers  in  any  wool- 
producing  state  of  the  country  ever  even 
saw  that  many  sheep  in  his  entire  life,  and, 
should  he  have  occasion  to  buy  them,  would 
for  a  moment  think  of  paying  for  them  by 
counting  out  663,395  ten-dollar  bills.  So 
long  as  such  problems  are  given  out  for 


the  consideration  of  pupils,  just  so  long  we 
may  expect  even  the  best  of  them  to  ask  the 
question  so  often  heard  in  our  algebra 
classes:  "What  is  all  of  this  'stuff'  good 
for,  anyway?" 

Contrast  with  this  problem  the  following, 
taken  at  random  from  an  algebra  recently 
published : 

"Two  boys,  A  and  B,  having  a  30-lb. 
weight  and  a  teeter  board,  proceed  to  de- 
termine their  respective  weights  as  follows : 
They  find  that  they  balance  when  B  is  6 
feet  and  A  5  feet  from  the  fulcrum.  If 
B  places  the  30-lb.  weight  on  the  board 
beside  him,  they  balance  when  B  is  4  and 


10 


SCIENCE 


A  is  5  feet  from  the  fulcrum.  How  heavy 
is  each  boy?" 

In  solving  this  problem  the  boy  has 
learned  just  as  much  mathematics  as  in 
solving  the  first.  In  addition,  his  mathe- 
matics has  been  brought  into  contact  with 
a  fundamental  physical  law,  and  incident- 
ally he  is  made  to  feel  that,  after  all,  his 
mathematics  is  of  consequence  to  him  in 
solving  the  sort  of  questions  in  which  he 
is  interested  or  is  likely  to  have  experience 
with  in  the  future. 

As  has  been  pointed  out,  a  change  in  the 
character  of  the  problems  is  gradually 
taking  place  in  our  mathematical  texts. 
Perhaps  a  word  of  caution  should  be  given 
lest  we  go  too  far  in  the  opposite  direc- 
tion, by  introducing  problems  which  re- 
quire a  technical  knowledge  and  experience 
beyond  the  comprehension  of  our  students. 
Perry's  calculus  is  a  conspicuous  illustra- 
tion of  this  danger.  The  subjects  discussed 
in  that  book  would  form  a  good  sequel  to  a 
certain  work  in  engineering,  but  the  book 
seems  to  be  hardly  suited  to  meet  the  needs 
of  American  schools  as  a  preparation  for 
engineering  study.  We  should  aim  to 
make  the  mathematical  work  practical  and 
in  harmony  with  engineering  practise,  but 
without  making  it  at  the  same  time  tech- 
nical in  its  applications,  or  without  going 
too  far  afield  by  teaching  mathematical 
physics. 

Another  improvement  which  has  recently 
become  noticeable  in  the  teaching  of  mathe- 
matics in  this  country  is  the  breaking  down 
of  the  traditional  barriers  between  the  dif- 
ferent branches  and  a  corresponding  closer 
correlation  of  the  different  subjects  in  the 
mathematical  curriculum.  In  several  of 
our  institutions  the  sharp  division  of 
freshman  work  into  algebra,  trigonometry, 
and  analytic  geometry  is  being  more  or 
less  disregarded  and  these  subjects  taught 
as  a  single  unit.      It  is  thought  that  the 


student  is  thus  enabled  to  grasp  more  read- 
ily these  subjects  as  a  whole,  and  that  the 
instructor  can  introduce  much  earlier  the 
principles  of  analytic  geometry  and  of  cal- 
culus and  postpone  to  the  later  part  of  the 
course  those  topics  which  are  relatively 
difficult  and  not  so  essential  to  the  elemen- 
tary work  of  the  course. 

This  plan  is  now  being  followed  some- 
what closely  at  the  University  of  Wiscon- 
sin. In  the  first  semester  fifteen  or  twenty 
recitations  are  devoted  to  the  elementary 
portions  of  trigonometry.  This  is  followed 
by  work  in  algebra,  including  the  theory 
of  complex  numbers,  using  trigonometry 
and  a  large  amount  of  graphic  work,  and 
the  elementary  principles  of  analytic  geom- 
etry. In  this  work  trigonometric  compu- 
tation and  the  use  of  the  slide  rule  form 
an  important  part.  In  the  second  semester 
the  algebra  and  trigonometry  are  continued 
and  combined  with  the  essentials  of  an- 
alytic geometry. 

This  correlation  of  the  work  of  the  fresh- 
man year  seems  to  have  been  most  thor- 
oughly worked  out  at  the  Massachusetts 
Institute  of  Technology,  where  Professors 
Woods  and  Bailey  have  recently  prepared 
a  text  covering  the  work  given  there  in  the 
freshman  year,  excluding,  however,  trig- 
onometry. The  indications  are  that  other 
institutions  are  also  contemplating  a  revi- 
sion and  better  correlation  of  the  work  of 
the  first  year. 

In  some  of  the  recent  books,  the  sharp 
division  of  the  calculus  into  differential 
and  integral  calculus  is  done  away  with, 
thus  making  it  possible  to  introduce  the 
student  to  a  wide  range  of  easy  applica- 
tions at  an  early  point  in  the  course  and 
to  relegate  to  its  proper  place  some  of  the 
more  difficult  parts  of  the  differential  cal- 
culus. There  is  a  tendency  also  to  intro- 
duce the  methods  of  the  calculus  earlier  and 
make  them  the  basis  of  portions  of  the  an- 


SCIENCE 


11 


alytie  geometry.  For  example,  Rose  Poly- 
technic Institute  gives  a  short  course  of 
thirty-six  recitations  in  the  subject  before 
analytic  geometry  is  taken,  and  what  is 
accomplished  there  in  this  formal  way  is 
undertaken  at  other  institutions  by  intro- 
ducing into  the  analytics  the  elementary 
notion  of  derivatives  or  by  teaching  the 
two  subjects  simultaneously. 

"While  all  are  agreed  that  for  engineers 
mechanics  should  stand  in  a  close  and  vital 
relation  to  the  calculus,  that  in  fact  it  is 
the  principal  reason  for  teaching  calculus, 
not  all  are  agreed,  however,  as  to  the  best 
method  of  accomplishing  this  purpose. 
Some  would  maintain  that  it  should  be 
taught  by  the  mathematical  department 
and  in  connection  with  calculus ;  others  and 
perhaps  the  larger  number  feel  that  it 
should  be  given  by  the  engineering  depart- 
ments and  made  to  follow  and  supplement 
the  calculus,  giving  the  student  his  first 
real  introduction  into  the  applications  of 
his  mathematics  to  the  fundamental  prin- 
ciples underlying  all  engineering  courses. 
However  this  may  be,  there  is  little 
doubt  that  more  applications  to  mechanics 
should  be  introduced  into  the  course  in 
calculus  than  is  now  usually  the  case,  even 
to  the  exclusion,  if  need  be,  of  some  of  the 
applications  to  geometry  frequently  given. 
Problems  in  work,  energy  and  stress  form 
just  as  legitimately  an  integral  part  of  a 
course  in  calculus  as  problems  in  order  of 
contact,  asymptotes  or  envelopes.  The 
applications  to  geometry  and  to  mechanics 
should  be  given  about  the  same  relative 
importance  in  a  well-balanced  course  in 
calculus. 

Descriptive  geometry  is  another  subject 
in  the  engineering  course  which  might  well 
be  revised  and  made  more  mathematical  in 
its  treatment.  It  is  to  be  regretted  that 
the  subject  has  in  this  country  degenerated 
into  little  more  than  mechanical  drawing. 


It  would  be  greatly  improved  for  engi- 
neers, as  well  as  the  general  student,  if  we 
should  inject  into  it  something  of  the  scien- 
tific spirit  given  it  in  European  schools. 

No  presentation  of  the  subject  under  dis- 
cussion would  be  complete  without  some 
consideration  of  the  preparation  which  the 
teacher  of  mathematics  has,  or  should  have, 
who  is  to  teach  the  subject  to  engineering 
students.  There  is  a  strong  feeling  in  some 
quarters  that  such  an  instructor  should  be 
a  trained  engineer  in  order  that  he  may  the 
better  appreciate  the  kind  of  applications 
which  are  best  suited  to  the  training  of  an 
engineer  and  to  make  sure  that  the  proper 
emphasis  be  placed  upon  those  topics  con- 
sidered essential  in  such  training.  Some 
would  go  still  farther  and  insist  that  even 
in  the  elementary  courses  in  mathematics 
usually  given  in  the  first  two  years,  the 
purpose  and  aim  of  the  prospective  engi- 
neer is  so  radically  different  from  that  of 
the  general  student  that  the  content  of  the 
course  itself  should  be  very  different  from 
what  is  best  suited  to  the  student  who 
elects  mathematics  as  a  part  in  a  general 
education. 

It  goes  without  saying  that  we  should 
eliminate  from  the  courses  for  engineering 
students  that  which  is  non-essential,  and 
we  should  make  them  as  practical  as  we 
may  by  the  generous  use  of  those  physical 
applications  which  will  give  the  students 
both  skill  and  facility  in  applying  mathe- 
matics to  such  concrete  cases  as  may  arise 
later  in  his  experience.  On  the  other  hand, 
it  would  be  disastrous  to  go  to  the  extent 
of  teaching  any  of  the  principles  of  mathe- 
matics empirically  or  of  permitting  stu- 
dents to  assume  as  already  established  for- 
mulas which  he  has  merely  to  learn  how  to 
apply.  We  should  avoid  the  danger  of 
going  too  far  in  allowing  the  student  to 
disregard  the  necessity  of  a  formal  demon- 
stration and  to  regard  lightly  the  logic  and 


12 


SCIENCE, 


the  philosophy  of  mathematics.  What  is 
needed  first  of  all  is  the  ability  on  the  part 
of  the  student  to  think  mathematically  and 
to  have  not  only  a  ready  but  an  intelligent 
command  of  the  fundamental  principles  of 
the  subject.  We  should  introduce  the  ap- 
plications of  mathematics  not  for  the  sole 
purpose  of  giving  the  student  a  foretaste  of 
the  things  which  are  in  store  for  him,  but 
because  such  applications  give  him  addi- 
tional opportunity  for  gaining  a  clearer 
comprehension  of  mathematical  processes 
and  principles  which  might  otherwise  be 
hazy;  and  I  wish  to  add  that  this  is  more 
essential  for  the  sound  training  of  the  spe- 
cial student  of  mathematics  with  his  limited 
opportunity  for  the  application  of  his  sub- 
ject to  physical  phenomena  than  it  is  for 
the  engineering  student  who  in  the  future 
is  to  have  opened  up  to  him  that  wide 
range  of  applications  which  his  technical 
studies  provide.  In  other  words,  what  is 
essential  in  the  way  of  applications  for  the 
engineering  student  in  the  first  two  years 
of  his  mathematical  work  gives  the  very 
best  training  for  the  student  who  is  taking 
mathematics  as  an  element  in  a  liberal  edu- 
cation. The  proper  place  for  differentia- 
tion, so  far  as  the  content  of  the  course  is 
concerned,  would  seem  to  be  after  the  com- 
pletion of  the  course  in  calculus  rather 
than  before.  I  present  this  as  a  plea  for 
the  general  student,  that  he  should  have 
more  of  the  applications  of  his  mathematics 
rather  than  that  the  engineering  student 
should  have  less.  Both  should  have  thor- 
ough drill  in  the  fundamental  principles 
of  the  subject  and  in  addition  all  of  the 
applications  of  those  principles  which  their 
limited  experience  and  knowledge  of  phys- 
ical phenomena  will  permit.  No  student, 
enginering  or  otherwise,  should  be  led  to 
regard  his  mathematical  work  in  the  same 
light  in  which  a  carpenter  may  properly 
regard  his  jack-plane,   a  mere  tool  with 


which  to  accomplish  certain  results ;  neither 
should  the  instructor  teach  mathematics  in 
the  spirit  in  which  a  skilled  operator  might 
regard  a  finely-equipped  machine  shop 
whose  sole  purpose  is  to  make  more  ma- 
chines. Both  extremes  are  to  be  avoided 
in  the  early  courses  in  mathematics.  The 
opportunity  for  specialization  and  differ- 
entiation should  come  later;  and  any  stu- 
dent who  is  not  capable  of  grasping  the 
fundamental  principles  of  the  mathematics 
usually  required  in  an  engineering  course 
should  not  aspire  to  a  bachelor's  degree 
from  a  large  university  or  technical  school. 
What  training  is  essential  or  desirable, 
then,  on  the  part  of  the  mathematical  in- 
structor of  engineering  students  to  best 
accomplish  the  general  results  here  set 
forth?  There  is  no  doubt  that  the  ideal 
thing  would  be  to  take  men  who  have 
completed  an  engineering  course  and  later 
supplemented  it  by  special  work  in  mathe- 
matics. This,  however,  does  not  seem  feas- 
ible because  of  the  few  who  could  be  in- 
duced to  take  such  a  course  of  training. 
It  would  be  quite  impossible  to  induce  a 
sufficient  number  of  engineers  to  take  up 
the  teaching  of  mathematics  to  meet  the 
demand,  even  if  that  seemed  desirable.  In 
most  cases  the  boy  enters  the  engineering 
course  with  the  view  of  practising  his  pro- 
fession when  he  has  completed  the  course. 
As  a  rule  he  has  little  taste  or  inclination 
for  teaching,  and  those  few  who  can  be  in- 
duced to  enter  the  less  remunerative  pro- 
fession of  teaching  are  absorbed,  as  indeed 
they  should  be,  by  the  engineering  depart- 
ments of  our  technical  schools.  To  put  an 
engineering  graduate  at  teaching  mathe- 
matics without  first  having  had  special 
training  in  mathematics  would  be  wholly 
undesirable.  Such  an  instructor  knows 
but  little  about  pure  mathematics  beyond 
the  elementary  courses  which  he  is  present- 
ing, and,  what  is  even  worse,  often  has  but 
little  interest.      If  he  can  be  induced  to 


SCIENCE 


13 


take  up  in  a  serious  way  the  study  of 
mathematics,  he  is  in  a  fair  way  of  becom- 
ing a  good  teacher  of  the  subject.  I  am 
thoroughly  convinced  that  mathematics 
should  be  taught  by  mathematicians  just 
as  engineering  should  be  taught  by  trained 
engineers;  but  the  mathematical  instructor 
who  wishes  to  teach  engineers  should  be 
familiar  with  the  general  field  of  applied 
mathematics— mechanics,  strength  of  ma- 
terials, thermodynamics,  and  in  addition 
so  much  of  the  broader  field  of  mathe- 
matical physics  as  possible. 

While  the  mathematical  instructor  should 
have  some  knowledge  of  its  applications,  it 
is  equally  desirable  that  the  teacher  of 
engineering  should  from  time  to  time  both 
refresh  and  revise  his  knowledge  of  the 
fundamental  things  in  mathematics,  to  the 
end  that  he  may  keep  his  methods  up  to 
date  and  adapt  his  teaching  to  the  kind  of 
mathematical  instruction  which  his  stu- 
dents have  had  and  avoid  those  methods 
and  those  forms  of  expression  which  have 
long  been  out  of  use. 

In  closing,  I  wish  to  add  that  the  rapid 
increase  in  engineering  students  has  so 
greatly  increased  the  demand  for  mathe- 
matical instructors  having  some  knowledge 
of  engineering  that  it  would  be  highly  de- 
sirable if  more  attention  should  be  paid  to 
the  preparation  of  men  for  such  positions. 
This  can  best  be  accomplished,  perhaps,  in 
those  universities  having  large  engineering 
departments  by  a  closer  correlating  of  the 
work  of  the  mathematical  department  with 
theoretical  work  in  engineering  and  mathe- 
matical physics.  It  is  to  be  regretted  that 
so  little  attention  in  this  country  is  now 
being  given  to  these  two  fields  of  mathe- 
matical activity.  Institutions  so  situated 
as  to  undertake  it  should  offer  to  its  stu- 
dents graduate  work  in  these  lines  in  every 
respect  worthy  of  a  doctor's  degree,  and 
likewise  to  its  instructors  both  opportunity 


and  encouragement  to  do  research  work  in 
this  broad  and  fruitful  field  of  human  en- 
deavor. Edgar  J.  Townsend 
University  of  Illinois 


THE    TEACHING    OF    MATHEMATICS    TO 

ENGINEERING   STUDENTS   IN 

FOREIGN    COUNTRIES1 

Your  committee  has  asked  me  to  speak 
of  the  teaching  of  mathematics  in  foreign 
engineering  colleges.  My  remarks  will 
have  reference  almost  exclusively  to  the 
German  colleges  and  schools,  partly  be- 
cause I  am  most  familiar  with  the  condi- 
tions existing  in  Germany  and  partly  on 
account  of  the  rather  instructive  campaign 
for  reforming  the  whole  teaching  of  mathe- 
matics, recently  inaugurated  in  Germany. 

As  regards  other  countries  I  will  only 
say  that  the  situation  in  England  and  Scot- 
land where,  during  the  last  quarter  of  a 
century,  technical  education  has  rapidly 
developed  on  quite  characteristic  and  indi- 
vidual lines,  deserves  careful  attention. 
But  I  am  not  sufficiently  well  acquainted 
with  the  facts  to  discuss  this  educational 
movement.  In  France,  it  is  well  known 
that  the  theoretical  training  given  to  engi- 
neers is  on  a  very  high  level,  higher  even 
than  in  Germany,  I  believe.  Thus,  the  re- 
quirements for  admission  to  the  Ecole 
Potytechnique,  or  even  to  the  Ecole  Cen- 
trale,  include  in  mathematics  almost  as 
much  as  our  engineering  students  get  in 
their  college  course.  On  the  top  of  this 
preparation,  the  student  receives  in  the 
Ecole  Polytechnique  an  excellent  two 
years'  course  in  higher  analysis  and  theo- 
retical mechanics,  and  then  only  is  he 
allowed  to  enter  upon  his  special  technical 
work.  It  must  also  be  taken  into  account 
that  admission  to  the  Ecole  Polytechnique 
is  by  competitive  examinations  held 
1  Read  before  Sections  A  and  D,  American  Asso- 
ciation for  the  Advancement  of  Science,  and  the 
Chicago  Section  of  the  American  Mathematical 
Society,  Chicago  meeting,  December  30,  1907. 


14 


SCIENCE 


throughout  France,  so  that  this  institution, 
receiving  as  it  does  the  pick  of  students 
from  the  whole  country,  can  maintain  a 
high  level  of  theoretical  excellency.  The 
Ecole  des  Ponts  et  Chaussees  and  the  Ecole 
des  Mines  to  which  the  student  passes 
from  the  Ecole  Poly  technique,  are  thus 
what  we  might  call  graduate  schools  of 
the  highest  rank. 

Turning  now  to  the  German  engineering 
colleges,  a  comparison  with  our  own  best 
engineering  colleges  shows  apparently  but 
little  difference,  both  as  regards  require- 
ments for  admission  and  as  to  the  schedule 
of  courses  offered  in  the  schools  themselves. 
Nevertheless,  I  believe  that  the  scientific 
standard  is  decidedly  higher  in  the  German 
than  in  the  American  engineering  college. 
I  am  not  here  concerned  with  the  question 
whether  such  a  high  standard  of  theoretical 
knowledge  is  essential,  or  even  desirable, 
for  the  engineer;  I  merely  state  the  fact. 
Moreover,  it  is  quite  possible  that  ulti- 
mately the  average  German  engineer  knows 
no  more  mathematics  than  the  average 
American  engineer.  All  I  wish  to  main- 
tain is  that,  in  my  opinion,  an  able  German 
student,  in  his  Technische  Hochschule,  or 
engineering  university,  can  gain  a  more 
thorough  scientific  equipment  than  an 
equally  able  American  student  in  his  alma 
mater. 

The  mathematical  requirements  for  ad- 
mission are  about  the  same  in  Germany  as 
with  us:  algebra,  geometry,  trigonometry. 
Not  a  few  students  now  enter  the  German 
engineering  college  with  some  knowledge 
of  analytic  geometry  and  even  of  calculus, 
but  many  still  come  without  this  knowl- 
edge. The  important  point  is  that  the  pre- 
paratory training  in  mathematics  (inclu- 
ding arithmetic)  is  distributed  systematic- 
ally and  continuously  over  a  period  of  nine 
years.  The  same  is  true  of  other  prepara- 
tory studies.     It  is  obviously  quite  impos- 


sible to  attain  in  a  four-year  high-school 
course  the  results  attained  in  the  nine-year 
course  of  a  German  Gymnasium,  Realgym- 
nasium,  or  Oberrealschule.  This  difference 
in  preparation  must  always  be  kept  in 
mind  in  making  comparisons  between  Ger- 
man and  American  universities. 

The  mathematical  courses  offered  in  the 
German  engineering  colleges  and  required 
for  a  degree  cover  plane  and  solid  analytic 
geometry,  differential  and  integral  calculus 
and  differential  equations— i.  e.,  about  the 
same  subjects  that  are  required  in  this 
country.  The  subject  of  theoretical  me- 
chanics, which  is  treated  rather  differently 
in  different  schools,  and  even  in  the  same 
school  for  different  degrees,  I  shall  here 
leave  out  of  consideration,  for  the  sake  of 
simplicity.  The  amount  of  time  devoted 
to  the  higher  mathematics,  not  including 
mechanics,  appears  roughly  from  the  fol- 
lowing table,  in  which  the  first  figure  in 
each  case  gives  the  number  of  hours  per 
week  devoted  to  lectures,  the  second  the 


Karlsruhe 

Stuttgart 

Munich 

Hannover...  

Danzig 

Braunschweig ... 
Zurich  


k 

u 

T3S 

0  B 

O  X) 

H 

pl,  a 

a>  B 

CI  Q) 

J  a 

V 

02  « 

~  o 

02 

02 

02 

02 

6+2 

6+2 

3 

2 

7+3 

6+4 

3+1 

6+3 

6+2 

5+2 

2+2 

8+1 

6+2 

6 

5 

4+1 

3+1 

8+2 

6+2 

2 

8+4 

8+4 

4+1 

17+4 
16+8 
19+9 
14+3 
18+2 
16+4 
20+9 


number  of  hours  devoted  to  "exercises." 
These  exercises  are  a  comparatively  recent 
innovation.  In  my  time  the  student  had 
nothing  but  lectures;  to  gain  a  working 
knowledge  of  the  subject  he  had  to  take  a 
text-book  and  work  for  himself.  Even 
now,  these  exercises  are  optional;  they 
probably  exist  everywhere,  although  the 
table  may  not  show  them.  There  are  no 
periodic  examinations  such  as  we  have  at 
the  end  of  each  semester;  but  most  stu- 
dents take  at  the  end  of  their  course  the 


SCIENCE 


15 


Staatsexamen,  or  if  particularly  ambitious, 
the  Diplomexamen.  The  lectures  in  mathe- 
matics are  rather  more  advanced  and  more 
complete  than  those  in  our  engineering  col- 
leges. But  the  requirements  in  the  final 
examinations  are  not  very  high. 

In  addition  to  the  more  thorough  prepa- 
ration of  the  German  student  and  to  the 
somewhat  higher  standard  of  the  lectures 
on  pure  mathematics,  and  largely  owing  to 
these  circumstances,  the  treatment  of  ap- 
plied mathematics  is,  I  believe,  on  a  higher 
level  in  Germany  than  in  this  country. 
The  student  is  better  prepared;  no  time  is 
lost  in  ' '  recitations, ' '  i.  e.,  in  trying  to  find 
out  whether  the  student  has  committed 
things  to  memory;  the  professor  is  thus 
enabled  to  treat  scientific  questions  scien- 
tifically. Besides,  on  an  average,  the  Ger- 
man professor  of  an  engineering  subject 
has  himself  a  higher  degree  of  scientific 
training  and  is  more  interested  in  the 
mathematical,  and  in  general  the  scientific, 
aspects  of  his  subject  than  his  American 
colleague. 

It  is  of  course  always  hazardous  and, 
moreover,  of  little  use  to  make  such  gen- 
eral statements  and  comparisons;  and  I 
do  not  wish  to  attach  any  great  importance 
to  them.  Neither  the  German  nor  the 
American  engineering  college  is  as  good  as 
it  might  be  or  should  be;  no  institution 
ever  is;  an  institution  is  good  only  in  so 
far  as  it  is  continually  changing,  develop- 
ing, rising.  The  above  comparisons  are, 
therefore,  given  merely  as  a  basis  for  better 
understanding  the  efforts  that  are  now 
made  in  Germany  for  the  improvement  of 
mathematical  teaching  in  all  its  phases.  To 
these  efforts  I  wish  to  call  your  special 
attention. 

The  German  movement  for  the  reform 
of  the  teaching  of  mathematics  is  of  a 
somewhat  complex  nature;  at  least  three 
different  movements  may  be  distinguished. 


One  of  these,  originating  with  the  German 
association  of  engineers  (Verein  Deutscher 
Ingenieure)  had  as  its  direct  object  the  im- 
provement of  the  mathematical  instruction 
in  the  engineering  colleges,  with  a  view  to 
making  the  instruction  less  abstract  and 
theoretical  and  more  practically  useful  to 
the  engineer.  To  a  certain  extent,  this 
object  has  been  attained.  Practical  exer- 
cises for  acquiring  a  working  knowledge 
of  mathematics  have  been  introduced  every- 
where, and  the  lectures  on  pure  mathe- 
matics have  become  less  theoretical.  Some 
of  the  originators  of  this  movement,  espe- 
cially Professor  Riedler,  of  the  Charlotten- 
burg  College,  went  so  far  as  to  demand  that 
in  engineering  colleges  mathematics  should 
be  taught  by  engineers.  Whether  or  not 
this  was  meant  as  more  than  a  threat  I  do 
not  undertake  to  say;  certainly,  as  far  as 
my  knowledge  goes,  no  attempt  has  ever 
been  made  in  a  German  engineering  college 
to  put  the  teaching  of  mathematics  in  the 
hands  of  any  one  but  a  trained  mathe- 
matician. But  I  believe  that  in  the  selec- 
tion of  men  for  such  positions  more  atten- 
tion has  been  paid  in  recent  years  to  the 
qualifications  of  the  aspirants;  mathe- 
maticians with  a  bent  towards  applied  sci- 
ence being  given  the  preference  for  posi- 
tions in  engineering  colleges. 

The  second  of  the  three  movements  re- 
ferred to  above  has  for  its  object  the  re- 
form of  the  teaching  of  mathematics  in  the 
universities.  It  is  the  oldest  of  these  move- 
ments, and  has  borne  fruit  in  a  variety  of 
ways.  But  I  can  here  only  advert  to  it 
very  briefly.  The  tremendous  creative 
mathematical  activity  that  characterized 
the  last  three  quarters  of  the  nineteenth 
century  in  Germany  led  to  a  condition  in 
the  universities  that  was  injurious  to  the 
preparation  of  teachers  for  the  secondary 
schools  (Gymnasium,  Realgymnasium, 
Oberrealschule) .     Too  much  stress  was  laid 


16 


SCIENCE 


on  leading  the  student  as  fast  as  possible 
to  original  research  in  some  special  line. 
The  system  has  been  described  as  a  system, 
not  of  double  entry,  but  of  double  for- 
getting; upon  entering  the  university  the 
students,  most  of  whom  are  fitting  for 
teaching  in  the  secondary  schools,  are  made 
to  forget  and  almost  despise  the  more  ele- 
mentary mathematics,  and  when  beginning 
their  professional  teaching  career  they  are 
again  compelled  to  forget  as  fast  as  possible 
all  the  higher  and  highest  mathematics  to 
which  they  had  devoted  most  of  their  time 
at  the  university.  The  remarkable  devel- 
opment of  mathematical  activity  in  our 
country  during  the  last  fifteen  or  twenty 
years  may  bring  about  a  similar  situation. 
Fortunately,  the  leaders  of  American  math- 
ematics are  well  aware  of  the  danger  of 
losing  the  healthy  contact  with  the  more 
elementary  mathematics  and  with  applied 
science.  Of  course,  it  is,  and  always  will 
be,  the  chief  object  of  a  real  university  to 
foster  original  research  and  productive 
scholarship.  But  it  is  well  even  for  the 
most  advanced  specialist  not  to  burn  the 
bridges  behind  him,  but  to  keep  in  mind 
the  connection  of  his  specialty  with  the 
foundations  of  knowledge,  on  the  one  hand, 
and  with  kindred  branches  of  science  on 
the  other.  As  Sir  Isaac  Newton  expressed 
it  in  his  quaint  way  in  a  letter  to  Dr.  Lord : 
"He  that  in  ye  mine  of  knowledge  deepest 
diggeth,  hath,  like  every  other  miner,  ye 
least  breathing  time,  and  must  sometimes 
at  least  come  to  terr,  alt  for  air." 

The  desire  to  make  the  university  teach- 
ing of  mathematics  more  practically  useful 
and  bring  it  into  live  contact,  as  far  as 
possible,  with  the  whole  tendency  of  mod- 
ern scientific  thought  led,  on  the  one  hand, 
to  a  strengthening  of  all  branches  of  ap- 
plied mathematics,  not  only  by  courses 
offered  in  the  universities,  but  also  by  such 
publications   as   the   Encyklopadie,   which 


includes  applied  mathematics  in  the  widest 
application  of  the  term ;  on  the  other,  it  led 
to  reforms  in  the  courses  offered  to  future 
teachers  of  mathematics,  and  ultimately  to 
a  thorough  investigation  of  the  teaching  of 
elementary  mathematics  in  the  secondary 
schools  of  Germany. 

The  improvement  of  the  teaching  of  ele- 
mentary mathematics  is  the  aim  of  the 
third  and  most  recent  mathematical  reform 
movement  in  Germany.  The  reforms 
proposed  in  this  connection  by  the  com- 
mittee of  the  German  Association  of  Natur- 
forscher  und  Aerzte,  at  the  Meran  meeting, 
in  1905,  appear  to  me  to  deserve  very  care- 
ful consideration.  They  would  apply,  in 
this  country,  to  the  teaching  of  mathe- 
matics not  only  in  the  high  schools,  but 
just  as  much  in  the  engineering  colleges. 
For,  with  the  preparation  that  our  students 
actually  have,  I  am  convinced  that  the  best 
method  of  imparting  a  good  working 
knowledge  of  the  elements  of  analytic 
geometry  and  calculus  is  not  through  lec- 
tures, but  through  actual  teaching  based 
mainly  on  solving  problems,  that  is,  by  the 
methods  not  of  the  German  university,  but 
of  the  German  secondary  school. 

The  proposals  of  the  committee2  do  not 
change  very  essentially  the  number  of 
hours  required  for  mathematics.  These 
are  to  be :  in  the  Gymnasium  as  well  as  in 
the  Realgymnasium,  four  hours  per  week 
in  each  of  the  nine  years ;  in  the  Oberreal- 
schule  generally  four  hours  per  week,  in 
the  third  and  fourth  years  six  hours.  The 
first  three  years  are  devoted  to  common 
arithmetic  and  intuitional  geometry,  the 
next  three  years  to  algebra  and  geometry 
carried  along  together,  the  last  three  years 
to  advanced  algebra,  trigonometry,  ad- 
vanced geometry,  conic  sections  (treated 
synthetically  and  analytically)  and,  in  the 

2  See  Zeitsohrift  fur  mathematischen  und  natur- 
wissenschaft  lichen  Unterricht,  Vol.  36  (1905), 
pp.  533-580. 


SCIENCE 


17 


Oberrealschule,  the  elements  of  the  cal- 
culus. Apart  from  matters  of  detail  this 
distribution  does  not  vary  very  much  from 
the  practise  now  followed  in  the  best  Prus- 
sian schools. 

While  thus  the  general  program  can  not 
be  said  to  constitute  a  radical  departure 
from  existing  conditions,  the  statement  of 
what  should  be  the  principal  aim  of  mathe- 
matical teaching  and  the  indications  given 
for  carrying  out  this  aim  throughout  the 
whole  course3  appear  to  me  as  the  most 
important  features  of  the  report.  In  addi- 
tion to  the  well-recognized  object  of  mathe- 
matical teaching  to  train  the  mind  in  rigor- 
ous logical  reasoning  the  report  insists  par- 
ticularly on  the  training  of  geometrical 
intuition  and  on  acquiring  the  habit  of 
functional  thinking.  The  carefully  pre- 
pared explanations  accompanying  the  de- 
tailed program  for  the  nine-year  course 
show  how  these  aims  should  guide  the  in- 
struction at  every  step.  The  insistence  on 
the  idea  of  the  functional  relation  can  not 
be  recommended  too  strongly  to  our  writers 
of  college  text-books,  from  trigonometry 
to  differential  equations.  But,  as  this  re- 
port demands,  it  should  even  enter  into  the 
very  elements  of  algebra  and  geometry. 

It  should  be  observed  that  the  committee 
that  prepared  this  report  was  not  composed 
of  mathematicians  only;  all  branches  of 
science  taught  in  the  secondary  schools 
were  represented  in  it;  and  all  these 
branches  received  equally  careful  attention. 
While  the  portion  of  the  report  devoted 
to  mathematics  covers  almost  the  whole 
range  of  the  subject,  from  arithmetic  to 
the  elements  of  the  calculus,  required  of 
our  engineering  students,  there  is  nowhere 
any  reference  to  students  of  engineering 
or  to  any  other  special  class  of  students. 
I  might,  therefore,  appear  out  of  order  in 
speaking  of  this  report  at  the  present  occa- 

"Loc.  tit.,  pp.  543-545,  550-553. 


sion.  But  I  wish  to  say  most  emphatically 
that,  in  my  opinion,  there  is  no  special 
"mathematics  for  engineers";  nor  is  there 
any  method  of  teaching  mathematics,  spe- 
cially adapted  to  engineering  students. 
If  it  is  wrong  to  present  mathematics  in  a 
form  so  abstract  as  to  make  it  unintel- 
ligible to  the  student,  it  is  just  as  wrong  to 
present  the  results  of  mathematics  in  a 
form  so  concrete  as  to  reduce  the  science  to 
a  mere  art  of  performing  certain  mechan- 
ical operations,  to  make  it,  as  the  saying 
goes,  a  mere  tool,  and  not  a  habit  of  think- 
ing. 

In  conclusion  allow  me  to  say  that  I 
should  be  the  last  to  advocate  a  remodeling 
of  our  institutions  of  learning  on  the  Ger- 
man plan,  or  the  French  plan,  or  any  other 
existing  plan.  But  I  believe  that  the  time 
has  come  in  this  country  when  one  or  two 
years  of  general  college  study  can  be  de- 
manded as  preparation  for  the  professional 
engineering  course,  at  least  for  those  more 
able  students  who  wish  to  obtain  a  thor- 
oughly scientific  preparation  for  their  pro- 
fessional career.  An  opportunity  should 
then  be  offered  to  students  of  engineering 
of  scientific  ability  to  extend  their  knowl- 
edge on  the  theoretical  side. 

Alexander  Ziwet 
University  of  Michigan 


THE   TEACHING  OF  MATHEMATICS  FOR 
ENGINEERS  * 

Mathematics,  from  the  standpoint  of 
the  engineer,  is  a  means,  and  not  an  end. 
It  is  an  instrument  or  tool  by  which  he  may 
determine  the  value  and  relations  of  forces 
and  materials. 

The  usefulness  of  tools  depends  upon 
the  sort  of  work  which  is  to  be  done,  upon 

1  Read  before  Sections  A  and  D  of  the  American 
Association  for  the  Advancement  of  Science  and 
the  Chicago  Section  of  the  American  Mathemat- 
ical Society,  at  the  Chicago  meeting,  December 
30,  1907. 


18 


SCIENCE 


the  kinds  of  tools  which  are  available  and 
upon  the  skill  of  the  man  who  uses  them. 
"We  may  inquire,  therefore,  what  are  the 
uses  to  which  the  engineer  may  apply 
mathematics?  What  kind  of  mathematics 
does  he  need?  And  what  skill  should  he 
possess  in  their  use? 

First,  then,  what  work  is  to  be  done  by 
the  young  men  who  are  now  taking  engi- 
neering courses?  A  few— and  only  a  few 
— will  be  original  investigators  or  de- 
signers who  will  need  mathematics  as  an 
instrument  of  research.  A  considerable 
number  will  regularly  employ  elementary 
mathematics  in  more  or  less  routine  cal- 
culations. Many  will  have  little  use  for 
mathematics,  as  engineering  courses  are 
recognized  as  affording  excellent  training 
for  various  business,  executive  and  other 
non-technical  positions,  particularly  in  con- 
nection with  manufacturing  and  operating 
companies.  It  has  been  stated  by  the  vice- 
president  of  a  large  electric  manufacturing 
company  that  not  over  ten  per  cent,  of 
the  technical  graduates  employed  by  that 
company  are  fitted  by  temperament  or  by 
education  to  take  up  with  success  the  work 
of  pure  engineering.  A  recent  classifica- 
tion of  the  graduates  of  Sibley  College, 
Cornell  University,  shows  that  about  half 
are  in  occupations  which  require  no  ad- 
vanced mathematics  and  it  is  probable 
that  many  of  the  SG'per  cent,  classed  as 
mechanical  and  electrical  engineers  seldom 
go  beyond  the  rules  of  arithmetic.  Hence 
a  goodly  proportion  of  engineering  gradu- 
ates do  not  need  to  be  mathematical  ex- 
perts. Their  mathematical  studies  need 
not  aim  to  produce  experts,  but  should 
have  as  a  principal  object  the  mathematical 
training  which  is  a  most  efficient  kind  of 
training  in  an  engineering  course.  On  the 
other  hand,  the  engineers  who  will  have 
practical  use  for  the  higher  mathematics 
will  find  their  ability  as  engineers  is  in  a 


large  measure  determined  by  their  ability 
as  mathematicians. 

Second,  the  question,  what  kinds  of 
mathematics  does  the  engineer  need?  is 
closely  related  to  the  class  of  work  he  is 
to  do.  In  general  a  great  deal  of  engineer- 
ing work  is  done  with  much  less  use  of 
higher  mathematics  than  most  professors 
probably  imagine ;  and,  furthermore,  it 
may  be  remarked,  with  much  less  than 
could  profitably  be  employed.  Engineers 
are  apt  to  use  ordinarily  the  mathematical 
methods  with  which  they  are  most  familiar 
and  which  will  bring  the  result  with  the 
least  effort.  One  man  employs  calculus, 
another  draws  a  diagram,  another  writes 
out  formula?,  while  another  gets  his  results 
by  mental  arithmetic.  The  object  is  to  get 
the  result. 

The  fundamental  idea  that  mathematics 
is  something  for  the  engineer  to  use  finds 
many  illustrative  analogies  in  ordinary 
tools.  Adaptation  is  the  first  requisite. 
Tools  should  be  suited  to  the  work  to  be 
done.  An  expensive  machine  tool  with  its 
refined  adjustments  is  quite  unnecessary 
for  executing  a  piece  of  work  which  can 
be  done  with  sufficient  accuracy  by  a  few 
minutes'  application  of  a  file.  An  ordi- 
nary calculating  slide  rule  is  infinitely 
better  than  a  table  of  seven-piece  logar- 
ithms in  every-day  work. 

On  the  other  hand,  it  is  particularly 
wasteful  to  attempt  to  execute  a  difficult 
and  intricate  piece  of  work  with  inade- 
quate tools.  But  more  important  than  the 
tool  is  the  skill  of  the  man  who  uses  it. 
A  skillful  workman  can  accomplish  results 
with  a  few  simple  tools  Avhich  others  can 
not  get  with  the  most  elaborate  special 
equipment. 

Third,  therefore,  skill  in  the  use  of 
mathematics  is  the  really  essential  thing. 
A  judicious  use  of  arithmetic  with  a  little 
algebra  or  a  simple  diagram  often  leads 
to   more  satisfactory   results   than   others 


SCIENCE 


19 


secure  through  elaborate  processes  involv- 
ing lengthy  equations  and  complicated 
operations.  In  the  latter,  errors  are  liable 
to  occur,  the  common-sense  import  of  the 
problem  is  apt  to  be  overlooked,  assump- 
tions may  be  made  to  facilitate  calculations 
which  are  physically  unwarranted  as  one 
loses  sight  of  the  physical  problem  in  the 
intricacy  of  the  mathematical  solution. 
Abstract  mathematical  studies,  if  pursued 
as  a  kind  of  intellectual  calisthenics,  may 
produce  a  pure  mathematician,  but  they 
may  unfit  a  man  for  practical  engineering. 
A  mathematician  is  not  necessarily  an 
engineer;  nor  is  an  elocutionist  necessarily 
a  good  lecturer,  nor  is  a  tool  expert  a  suc- 
cessful manufacturer. 

Mathematics  is  used  in  engineering  to 
express  the  quantitative  relations  of  nat- 
ural phenomena.  The  mathematician  de- 
lights in  the  relations:  he  divorces  them 
from  the  phenomena  and  gives  them  ab- 
stract expression,  while  the  engineer  is  con- 
cerned with  the  natural  phenomena;  he 
demands  the  physical  conception;  the  me- 
dium of  expressing  these  relations  is  of 
secondary  consequence. 

The  mathematician  evolves  the  equation 
for  a  parabola  and  finds  a  convenient 
illustration  in  the  law  of  projectiles.  The 
engineer  finds  that  a  physical  result  fol- 
lows from  the  application  of  certain  forces, 
and  uses  the  formula  merely  as  a  con- 
venient method  of  expressing  the  law.  The 
analogue  in  the  case  of  mechanical  tools  is 
found  by  regarding  a  set  of  drawing  in- 
struments or  a  transit  or  a  lathe,  as  some- 
thing intelligently  designed,  properly  pro- 
portioned, accurately  made  and  finely 
finished,  the  merit  of  which  lies  in  its  own 
inherent  excellence;  or,  on  the  other  hand, 
by  considering  them  as  tools  adapted  for 
doing  a  certain  range  and  character  of 
work  with  a  sufficient  degree  of  accuracy 
and  at  low  cost. 

A  manual-training  school  gives  familiar- 


ity with  mechanical  tools  and  mathe- 
matical study  gives  familiarity  with  in- 
tellectual tools.  In  work  with  the  manual 
tool  the  boy  uses  it  for  making  something 
— he  learns  the  principle  on  which  it 
operates  and  the  way  to  use  it,  by  making 
something;  if  it  is  something  useful  it 
awakens  a  higher  interest  than  does  some 
fancy  device.  Likewise  training  of  engi- 
neers in  mathematics  should  be  by  doing 
something,  by  the  solving  of  problems,  by 
dealing  with  real  rather  than  abstract  con- 
ditions. Let  this  training  be  secured  while 
applying  mathematics  to  its  normal  and 
legitimate  purpose  as  an  auxiliary  in  the 
study  of  other  branches. 

In  the  teaching  of  mathematics  for  its 
own  sake  stress  is  apt  to  be  laid  upon  the 
processes  of  deriving  results  rather  than 
the  real  meaning  of  the  results  themselves. 
An  engineer  who  uses  logarithms  has  no 
more  concern  regarding  their  derivation 
than  the  ordinary  user  of  the  dictionary 
for  finding  the  pronunciation  of  words  has 
in  their  etymological  derivation.  The 
ability  to  reproduce  demonstrations  in 
higher  mathematics  from  memory  with  the 
book  shut  is  often  not  as  important  as  it 
is  to  understand  them  with  the  book  open. 
In  general  an  engineer,  who  has  occasion  to 
use  higher  mathematics,  will  not  be  inter- 
ested in  evolving  difficult  equations,  nor 
will  he  appeal  to  his  memory,  but  with 
text-book  or  reference  before  him  he  will 
seek  the  things  he  wants  to  use.  He  should 
know  where  to  find  them  and  how  to  use 
them. 

In  emphasizing  what  a  skilled  mechanic 
can  make  with  very  ordinary  tools,  or  the 
true  engineer  can  accomplish  with  the 
parallelogram  of  forces  and  the  rule  of 
three,  there  is  no  intention  of  discrediting 
the  value  of  fine  equipments,  either  me- 
chanical or  mathematical,  if  there  be  the 
ability  to  use  them. 

Possibly  the  practical  utility  of  mathe- 


20 


SCIENCE 


matics  may  appear  to  be  urged  too 
strongly,  particularly  as  the  writer  really 
believes  in  thorough  mathematical  train- 
ing, but  he  has  seen  so  many  cases  in  which 
mathematical  instruction  has  never  been 
digested  and  assimilated,  he  has  seen  simple 
problems  confused  by  unnecessary  mathe- 
matical complications,  he  has  seen  men 
satisfied  with  results  which  are  absurd 
because  of  some  mathematical  equations — 
sometimes  quite  unnecessary — which  seem 
to  obliterate  common-sense  perspective, 
and  he  recalls  the  new  insight  into  mathe- 
matics which  came  through  "Analytic 
Mechanics"  under  Professor  S.  W.  Eobin- 
son  at  the  Ohio  State  University,  and 
''Problems  in  Mechanics,"  under  Dr. 
Fabian  Franklin  at  Johns  Hopkins 
University,  that  he  feels  there  is  little 
danger  in  over-emphasizing  the  importance 
of  concrete  training  in  mathematical 
study.2 

The  practical  questions  which  the  dis- 
cussion of  this  subject  presents  are  these: 

What  mathematical  subject-matter  should 
be  covered?     And, 

How  should  it  be  taught? 

The  first  difficulty  is  that  there  is  not, 
and  can  not  be,  a  differentiation  in  tech- 
nical education  which  is  at  all  comparable 
with  the  wide  range  of  occupations  into 
which  graduates  will  enter.  We  may  as- 
sume, therefore,  that  we  are  considering  the 
case  of  the  average  engineering  student, 
taking  for  granted  that  options  may  be 
used  by  the  best  students  for  enabling 
them  to  take  up  the  more  advanced  and 
difficult  mathematics.  Obviously  the  stu- 
dent should  have  enough  mathematics  to 
enable  him  to  demonstrate  the  important 
engineering  laws  and  formulas  and  to  read 
intelligently  mathematically  written  engi- 
neering literature.     While  only  the  rela- 

2  Both  of  these  teachers  of  mathematics  had 
been  trained  as  engineers  and  had  practised  the 
profession. 


tively  simple  mathematics  is  commonly 
used  by  engineers,  yet  the  ability  to  handle 
new  problems  with  confidence  requires  a 
thorough  understanding  and  appreciation 
of  the  significance  of  the  mathematical  and 
physical  basis  of  the  laws  and  phenomena 
he  is  to  use.  A  man  who  is  a  thorough 
mathematician  and  knows  how  to  apply 
his  knowledge  has  a  great  advantage  over 
the  pure  mathematician  or  the  man  with- 
out mathematical  equipment.  The  better 
knowledge  one  has  of  the  complex,  the 
more  certainty  he  has  in  applying  the 
simple.  A  student  should  understand 
something  of  the  power  of  the  advanced 
mathematics  and  the  field  of  its  effi- 
cient application.  Although  he  may 
not  be  expert  in  using  it  himself,  he 
will  know  when  to  call  for  a  mathematical 
expert. 

An  engineer  of  fairly  wide  experience 
remarked  a  short  time  ago :  ' '  The  ordinary 
engineer  does  not  use  higher  mathematics 
because  he  doesn't  know  how.  He  does 
not  have  the  proper  conception  of  the 
fundamental  principles  of  the  calculus  be- 
cause the  subject  has  been  taught  by  men 
whose  ideals  are  those  of  pure  mathe- 
matics. ' ' 

If  mathematics  is  something  for  engi- 
neers to  use,  let  its  use  be  taught  to  engi- 
neering students.  After  the  fundamentals 
are  learned,  the  students  should  attack  the 
engineering  problem  at  once  and  bring  in 
mathematics  as  a  means  of  solving  it. 
Mathematics  is  often  advocated  for  de- 
veloping the  reasoning  powers  and  the 
ability  to  reason  from  cause  to  effect. 
There  is  danger,  however,  that  mathe- 
matical machinery  may  make  the  mere 
process  obscure  the  cause  and  the  effect. 
Let  them  be  foremost,  with  the  process 
secondary  or  auxiliary  to  them. 

The  way  mathematics  is  brought  to  bear 
on  some  engineering  problems  reminds  one 
of  the  story  of  the  old  lady  who  greatly 


SCIENCE 


21 


admired  her  preacher  because  he  could 
take  a  simple  text  and  make  it  so  very- 
complicated. 

Old  traditions  have  not  wholly  disap- 
peared, the  fear  of  degrading  the  pure 
science  of  mathematics  by  applying  it  to 
useful  things  still  lingers— in  influence,  if 
not  in  precept.  We  must  go  further  and 
adapt  mathematics  to  engineering,  not  only 
in  subject  matter,  but  in  method.  A 
mathematical  teacher  with  no  patience  for 
anything  except  mathematics  will  probably 
teach  a  kind  of  mathematics  which  has  no 
connection  with  anything  except  mathe- 
matics. Engineering  mathematics  may  be 
better  taught  as  a  part  of  engineering  by 
an  engineer,  than  as  a  part  of  mathe- 
matics by  a  pure  mathematician.  The 
marker  of  levels  and  transits  who  is  expert 
in  the  construction  of  the  instruments  and 
an  enthusiast  over  the  accuracy  of  the  sur- 
faces, the  excellence  of  the  bearings,  the 
near  approach  to  perfection  in  the  gradua- 
tion and  the  general  refinement  and  beauty 
of  workmanship,  may  make  a  good  in- 
structor on  instruments,  but  a  poor  teacher 
of  civil  engineering. 

After  all,  it  is  not  so  much  abstract 
courses  as  it  is  personal  men  with  which  we 
have  to  do,  it  is  not  mere  knowledge  of 
facts  or  facility  in  mathematical  manipula- 
tion, but  it  is  training.  The  young  man 
is  to  be  developed,  his  native  individuality 
is  to  be  the  basis,  he  is  to  increase  not  only 
his  knowledge,  but  his  powers  and  the 
ability  to  use  them.  It  is  not  mathematical 
skill  so  much  as  a  mathematical  sense,  or 
mathematical  common-sense,  which  is 
wanted.  With  pure  mathematics  as  a  sci- 
ence we  have  no  quarrel— and  little  affilia- 
tion. 

If  you  ask  men  who  use  engineering 
graduates  what  qualities  they  should  pos- 
sess, you  will  find  that  special  prominence 
is  given  to  ' ' common-sense "  and  "the 
ability   to   do  things."     In  mathematical 


training  it  is  quality  rather  than  quantity 
which  is  of  first  consequence.  It  should 
develop  the  facility  for  systematic  and 
logical  reasoning,  thus  furnishing  a  gen- 
eral method  as  well  as  a  specific  means  of 
getting  results. 

We  are  concerned  with  applied  mathe- 
matics. The  ability  to  state  a  problem ;  to 
recognize  the  elements  which  enter  into  it; 
to  see  the  whole  problem  without  over- 
looking some  important  factor ;  to  use  good 
judgment  as  to  the  reliability  or  accuracy 
of  the  data  or  measurements  which  are  in- 
volved ;  and,  on  the  other  hand,  the  ability 
to  interpret  the  result;  to  recognize  its 
physical  significance;  to  get  a  common- 
sense  perspective  view  of  its  meaning  and 
the  consequences  which  may  follow;  to 
note  the  bearing  of  the  various  data  upon 
the  final  result ;  to  determine  what  changes 
in  original  conditions  may  change  a  bad 
result  into  one  which  is  practical  and  effi- 
cient—such abilities  as  these  are  of  a 
higher  order  than  the  ability  to  take  a 
stated  problem  and  work  out  the  answer. 
It  may  be  urged  that  all  this  is  not  strictly 
mathematics.  But  it  is  just  this  sort  of 
judgment  and  insight  which  makes  mathe- 
matics really  useful,  and  without  them 
there  is  danger  that  they  may  be  neither 
safe  nor  sane. 

The  trend  in  education  is  to  a  closer  re- 
lation to  the  affairs  of  life.  Science  and 
applied  science,  scientific  and  engineering 
laboratories,  are  overcoming  old  ideas  and 
prejudices.  Modern  engineering  develop- 
ment brings  its  transforming  influence  to 
bear  upon  education  as  well  as  the  utilities 
of  modern  life.  The  engineering  school 
has  had  a  phenomenal  growth  within  the 
lifetime  of  the  recent  graduate— a  growth 
in  ideals  and  methods  as  well  as  students 
and  equipment.  It  has  raised  and  agitated 
broad  questions  as  to  what  constitutes 
efficient  education  for  producing  effective 
men.     It  has  aimed  to  combine  not  only 


22 


SCIENCE 


the  abstract  with  the  concrete,  the  lecture 
room  with  the  laboratory,  and  the  scientific 
experiment  with  the  practical  test;  but  it 
has  sought  by  various  means  to  bring  the 
work  of  the  school  into  close  relation  with 
active  professional  and  commercial  prac- 
tise. It  has  a  definiteness  of  aim  and  pur- 
pose which  other  educational  courses  are 
apt  to  lack.  It  sets  out  to  produce  men 
who  can  deal  with  forces  and  materials 
according  to  scientific  principles.  It  de- 
velops men  whose  contact  with  physical 
facts  and  natural  laws  are  first  hand  and 
whose  ability  to  reason  logically  fit  them 
for  dealing  with  new  problems.  The 
training  which  fits  men  for  handling  engi- 
neering problems  is  the  kind  that  is  needed 
for  dealing  with  the  organization  and  di- 
recting of  men.  The  sphere  of  the  engi- 
neer is  one  the  scope  of  which  will  con- 
tinue to  increase  as  engineering  education 
and  training  produce  men  whose  contact 
with  natural  phenomena  gives  them  an  in- 
herent respect  for  facts  as  their  premises, 
who  are  able  to  think  straight  to  logical 
and  common-sense  conclusions,  who  have 
an  equipment  of  technical  knowledge  and 
who  can  produce  results. 

In  discussing  the  teaching  of  mathe- 
matics to  engineers,  we  should  emphasize 
not  the  mathematics  nor  the  engineers,  but 
the  teaching.  Aside  from  the  imparting 
of  knowledge  and  technical  ability,  the 
teaching  of  mathematics  gives  opportunity 
for  training  in  the  use  of  logical  methods 
and  in  the  drawing  of  intelligent  conclu- 
sions from  unorganized  data  which  will 
make  efficient  men,  whether  they  follow 
pure  engineering,  or  semi-technical,  or 
business  pursuits.  Such  teaching  does  not 
come  from  the  text-book;  it  must  be  per- 
sonal—it comes  from  the  teacher.  He 
must  be  in  sympathy  with  engineering 
work  and  have  a  just  appreciation  of  its 
problems  and  its  methods.     He  must  be 


imbued  with  the  spirit  and  the  ideals  of 
the  engineer. 

Chas.  F.  Scott 

Pittsburgh,  Pa. 


THE   POINT    OF    VIEW   IN    TEACHING 
ENGINEERING   MATHEMATICS  » 

I  hardly  know  why  I  should  have  been 
asked  to  address  you  at  this  conference. 
Possibly,  however,  the  fact  that  I  am  a 
civil  engineer  by  profession,  without  hav- 
ing been  permitted  ever  to  practise  this 
profession,  and  the  additional  fact  that  I 
have  been  a  professional  teacher  of  mathe- 
matical physics,  without  having  been  per- 
mitted to  continue  in  this  work,  have  led 
your  committee  to  think  that  I  might 
furnish  a  conspicuous  illustration  of  the 
failures  to  which  colleges  and  universi- 
ties may  lead  in  these  lines  of  endeavor. 

Having  listened  attentively  to  the  three 
formal  papers  just  read,  I  find  it  essential 
to  revise  my  program  and  instead  of  fol- 
lowing similar  lines  to  those  of  the  preced- 
ing speakers,  it  seems  essential  to  take 
direct  issue  with  them.  This  I  am  dis- 
posed to  do,  not  so  much  because  I  differ 
wholly  from  the  views  they  have  set  forth, 
as  because  it  seems  necessary  to  have  other 
sides  of  the  questions  they  have  discussed 
represented.  The  preceding  speakers  ap- 
pear to  me  to  have  taken  themselves  some- 
what too  seriously.  This  is  a  general 
fault  of  both  theoretical  and  practical  edu- 
cationalists. My  own  experience  leads  me 
to  conclude  that  in  educational  affairs  the 
teacher,  the  school,  the  college  and  the 
university  play  a  much  less  important  role 
than  we  commonly  suppose.  In  fact,  I 
have  reached  the  provisional  conclusion 
that  the  majority  of  our  students  turn  out 
fairly  well  in  the  world  not  so  much  by 

1  Extempore  remarks  before  Sections  A  and  D 
of  the  American  Association  for  the  Advancement 
of  Science  and  the  Chicago  Section  of  the  Amer- 
ican Mathematical  Society,  at  the  Chicago  meet- 
ing, December  30,  1907. 


SCIENCE 


23 


reason   of  the  academic   instruction  they 
receive  as  in  spite  of  it. 

My  impression  also  is  that  in  taking  our- 
selves too  seriously  as  teachers  of  one  sub- 
ject or  another,  we  have,  as  a  rule,  quite 
underestimated  the  magnitude  and  the  dif- 
ficulty of  the  psychological  problems  with 
which  we  have  to  deal.  We  have,  as  a 
rule,  quite  overestimated  the  capacity  of 
our  average  student,  and  have  thus 
usually  expected  too  much  from  him.  It 
is,  of  course,  desirable  to  set  our  ideal 
high  and  try  to  rise  to  an  elevated  in- 
tellectual level;  but  in  doing  so  we  have 
commonly  neglected  the  influence  of 
heredity  as  well  as  of  environment.  I  am 
inclined  to  think  Dr.  Holmes  was  right 
when  he  said  that  it  is  essential  in  the 
generation  of  a  gentleman  to  begin  four 
hundred  years  before  he  is  born.  So  also 
is  it  necessary,  if  we  wish  to  develop  a 
student  into  a  first-class  scholar,  to  begin 
back  some  generations  before  we  take  up 
the  formal  work  of  training  in  our  col- 
leges or  schools  of  engineering.  It  is  an 
important  fact,  also  too  commonly  over- 
looked, that  the  fundamental  ideas  in- 
volved in  the  mathematics  and  in  the 
mathematical  physics  essential  to  the  pre- 
liminary training  of  a  prospective  engi- 
neer are  far  more  difficult  of  compre- 
hension than  we  are  wont  to  suppose.  As 
a  rule,  I  think  we  begin  our  elementary 
mathematics  somewhat  too  early  for  the 
average  mind.  The  result  is  that  our  stu- 
dents acquire  a  mere  literary  knowledge 
of  the  subject  without  grasping  the  basic 
ideas  essential  to  clear  thought  and  espe- 
cially essential  to  applications.  I  am  go- 
ing to  give  you  some  illustrations  of  this 
fact.  They  will  show  how  difficult  it  is 
for  the  average  mind  to  attain  a  proper 
understanding  of  mathematico-physical 
concepts.  The  difficulties  here  are  much 
the  same  as  the  difficulties  of  grammar. 
As  you  know,  children  learn  to  speak,  and 


often  speak  very  well,  long  before  they 
know  anything  of  formal  grammar,  and 
this  is  the  natural  mode  of  development, 
for  the  logic  and  subtleties  of  grammar 
can  be  appreciated  only  by  rather  mature 
minds. 

But  if  the  concepts  which  belong  to  the 
study  of  language  and  of  grammar  are 
rather  formidable,  those  which  belong  to 
the  higher  mathematics  and  mathematical 
physics  are  profoundly  more  difficult  of 
adequate  comprehension.  Let  me  illus- 
trate this  point  by  a  citation  from  experi- 
ence furnished  by  the  case  of  a  graduate 
from  one  of  our  universities  who  pre- 
sented himself  to  me  a  few  years  ago,  while 
I  was  dean  of  a  graduate  school  of  Co- 
lumbia University,  as  a  candidate  for  a 
higher  degree  in  mathematical  physics. 
This  student  had  studied  mechanics  and 
had  attained  a  degree  in  engineering.  In 
order  to  learn  something  of  the  breadth 
and  depth  of  his  knowledge,  I  asked  him 
what  it  is  that  makes  the  trolley  car  run 
after  the  current  is  cut  off.  He  answered, 
"It  is  the  force  of  the  momentum  of  the 
power  of  the  energy  of  the  car. ' '  There  is 
no  reason  to  suppose  that  he  had  not  re- 
ceived good  mathematical  and  physical 
training,  and  yet  it  is  plain  from  the 
answer  he  gave  me  that  he  knew  next  to 
nothing  of  the  meaning  of  the  terms  he 
used.  I  may  cite  another  case  of  a  suc- 
cessful practising  engineer,  who  was  a 
pupil  of  no  less  authorities  in  mechanics 
and  engineering  than  Lord  Kelvin  and 
Rankine.  This  man  wrote  me  a  letter  in 
which  he  sought  to  convince  me  that 
Newton  and  his  followers  are  all  wrong 
with  regard  to  the  parallelogram  of  im- 
pulses. "Thus,"  he  said  in  his  letter,  "if 
a  particle  starts  out  from  a  given  point 
under  the  simultaneous  action  of  two  im- 
pulses, it  will  not  move  in  the  parallelo- 
gram of  the  impulses,  but  it  will  move  in  a 


24 


SCIENCE 


tautochronous,  brachistochronic,  plane  cate- 
nary curve  of  a  resilient  character." 

These  illustrations  show  how  extremely 
difficult  it  is  to  master  the  fundamental 
ideas  which  belong  to  a  great  science ;  and 
the  difficulties  are  so  great  that  I  am  dis- 
posed to  excuse,  or  at  any  rate  palliate,  the 
blunders  made  by  our  average  student. 
He  is,  in  fact,  with  all  his  blunders,  not 
very  far  behind  many  of  his  teachers,  for 
it  is  not  uncommon  for  them  to  use  in 
their  lectures  and  text-books  words  not  at 
all  free  from  ambiguity.  Witness,  in  fact, 
the  loose  use  of  such  words  as  force,  power, 
pressure,  stress,  and  strain  in  some  of  the 
best  text-books  and  treatises  of  the  nine- 
teenth century.  The  word  ''power,"  for 
example,  is  often  used  in  two  radically 
different  senses  in  the  same  sentence. 

These  difficulties  and  ambiguities  lead 
me  to  suggest,  in  opposition  to  the  precepts 
laid  down  by  a  previous  speaker,  that  we 
may  well  consider  the  desirability  of  print- 
ing mathematical  books  free  from  demon- 
strations but  containing  plain  statements 
of  facts.  I  have  used  such  books  myself 
and  am  disposed  to  think  they  are  amongst 
the  best  books  we  may  place  in  the  hands 
of  a  student.  The  simple  fact  is  that  we 
do  not  follow  a  logical  order  of  develop- 
ment in  acquiring  knowledge.  We  pro- 
ceed rather  by  the  method  of  "trial  and 
error,"  and  we  often  find  out  the  facts 
with  regard  to  an  item  of  learning  long 
before  we  become  aware  of  the  principle 
involved. 

Hence  I  think  the  reason  why  few  of  our 
engineers  know  much  about  the  formalities 
of  mathematics  and  mathematical  physics 
after  they  get  through  college  is  plain 
enough.  They  are  driven  over  so  many 
subjects  during  the  four  years  of  their 
college  life  that  they  have  little  or  no  time 
for  reflection.  This  latter  must  come  later 
in  life  when  the  mind  has  developed  a  suf- 
ficient  degree   of  maturity  to   appreciate 


the  more  recondite  principles  which  lie  at 
the  foundation  of  all  the  higher  learning. 
This  fact  is  well  illustrated  also  by  the 
case  of  our  friends,  the  humanists,  who 
have,  as  you  know,  for  a  long  time  pro- 
posed the  study  of  geometry  for  "mental 
discipline."  As  a  matter  of  fact,  those 
who  have  acquired  anything  like  a  grasp 
of  geometrical  principles  known  that  very 
few  students  of  Euclidean  geometry  ac- 
quire anything  like  an  adequate  appre- 
ciation of  the  ideas  involved,  and  it 
is  only  in  the  rarest  instances  that  these 
students  pursue  the  subject  after  leaving 
college. 

I  have  not  much  sympathy  with  the 
engineers  who  would  like  to  have  their 
own  kind  of  mathematics,  and  I  am  not 
disposed  to  commend  very  highly  the 
works  on  calculus  and  other  branches  of 
pure  mathematics  designed  especially  for 
engineers.  On  the  other  hand,  our  modern 
mathematicians  have  generally  failed  to 
understand  the  needs  of  the  engineer. 
Our  more  recent  type  of  mathematician 
has  devoted  himself  too  largely  to  the  re- 
fined questions  of  convergence  and  diverg- 
ence of  series  and  of  existence  theorems  to 
properly  equip  him  for  the  numerous  and 
important  applications  which  the  ideal 
engineer  should  be  able  to  make  of  his 
mathematical  knowledge.  The  modern 
mathematician  seems  prone  to  make  the 
engineer  with  some  degree  of  mathematical 
talent  afraid  of  himself.  I  have  met  some 
students  whose  early  training  had  filled 
them  with  caution  to  such  a  degree  that 
they  would  not  use  infinite  series  for  fear 
that  a  divergent  one  might  be  encountered. 
It  is  known,  however,  as  a  matter  of  fact, 
that  most  series  essential  in  the  applica- 
tions of  mathematics  to  mathematical 
physics  are  safe  in  this  regard,  and  one 
of  the  best  ways  for  the  elementary  student 
to  learn  of  the  degree  of  convergence  is 


SCIENCE 


25 


to  apply  numerical  computation  to  these 
series. 

This  leads  me  to  say  a  few  words  con- 
cerning numerical  computations,  in  which 
very  few  engineers  and  still  fewer  mathe- 
maticians show  any  degree  of  proficiency. 
It  seems  to  me  this  is  one  of  the  most 
lamentable  defects  of  our  elementary  teach- 
ing in  mathematics,  though  here  as  else- 
where the  intrinsic  difficulties  are  much 
greater  than  we  commonly  suppose.  This 
fact  is  in  evidence  at  almost  every  meeting 
of  our  scientific  societies,  for  it  oftenest 
happens  that  the  author  of  a  paper  involv- 
ing numerical  calculation  will  talk  of  the 
decimals  involved  instead  of  the  significant 
figures.  Thus,  he  will  say,  "this  result  is 
correct  to  five  places  of  decimals,"  when 
he  should  say,  "this  result  is  correct  to  a 
specified  number  of  significant  figures," 
the  latter  form  of  expression  being  requi- 
site to  indicate  the  degree  of  precision  at- 
tained. There  is  a  grave  defect  in  our 
elementary  teaching  in  these  matters;  but 
it  arises  from  the  fact  that  almost  none  of 
our  teachers  of  elementary  mathematics 
are  qualified  to  understand  the  refinements 
and  the  difficulties  of  precision  in  compu- 
tation. Thus,  it  often  happens  that  stu- 
dents will  give  results  to  five  or  seven 
significant  figures  when  the  data  do  not 
justify  any  such  apparent  precision. 

To  correct  these  evils  we  must  have  a 
convention  of  mathematicians,  engineers 
and  professional  computers  who  will  show 
authors  how  to  produce  elementary  text- 
books giving  adequate  attention  to  these 
matters. 

As  regards  numerical  computation,  there 
is  in  general  need  of  more  practise,  since 
it  is  through  the  concrete  that  we  learn  of 
the  abstract  and  the  fundamental.  No  im- 
portant formula  in  any  text-book  or 
treatise  should  go  without  an  appropriate 
illustrative  numerical  example. 

I  would  like  to  take  advantage  of  this 


occasion  to  express  a  hope  with  regard  to 
the  future  of  our  country  and  to  the  possi- 
bility   of    development   which   may    come 
through     suitable      cooperation     between 
mathematicians    and    engineers.      Nothing 
delights  me  more  than  to  attend  a  meeting 
of   this   kind   where    mathematicians    and 
engineers  have  come  together.      It  is  an 
auspicious  sign  of  the  times.     It  is  one  of 
the  results  I  have  been  looking  forward  to 
for  the  past  thirty  or  forty  years.     Some 
of  us  here  are  old  enough  to  have  lived  in 
two  epochs,  namely,  the  pre-scientific  and 
the  present  epoch.      We  can  remember  a 
time  when  engineers  could  not  have  got  a 
hearing  such   as  they  have  to-day.     The 
history  of  their  rise  and  development,  at 
least  in  this  country,  is  well  known  to  some 
of  us.     It  dates  back  to  a  time  only  about 
forty   years   ago.     During   this   time   the 
engineers  have  fought  their  way  forward 
to  the  position  now  accorded  them  in  con- 
temporary society.     They  have  won  a  place 
in  public  esteem  without  which  it  would 
have  been  impossible  to  hold  such  a  con- 
ference as  we  are  holding  to-day.     This 
esteem  has  been  won  in  spite  of  much  op- 
position, coming  especially  from  the  older 
academic  institutions;  but  now  having  at- 
tained  adequate  recognition  especially  as 
practising    engineers,    we    have    a    much 
higher  duty  to  perform,  and  this  I  trust  we 
shall  be  able  to  meet  adequately  through 
cooperation    with    our    friends    the  pure 
mathematicians.     I  know  of  no  work  more 
important  to  the  general  advancement  of 
mathematico-physical    science    than    that 
which   may   lead   to    the   development   of 
mathematical  physicists,  men  who  possess 
at  once  good  mathematical  knowledge  and 
correspondingly    adequate    equipment    in 
physical  science.      Here  is  a  field  greatly 
in  need  of  concentrated  effort  and  of  ade- 
quate   appreciation.      It   is   a    lamentable 
fact  that  while  we  can  easily  develop  pure 
mathematicians  of  a  high  order  and  experi- 


26 


SCIENCE 


mental  physicists  of  an  equally  high  order, 
it  seems  very  difficult  for  us  to  develop 
minds  possessing  both  qualities.  To  a 
large  extent  I  thing  the  development  of 
pure  mathematics  in  the  future  will  de- 
pend, as  in  the  past,  on  the  stimulus  fur- 
nished by  mathematico-physical  ideas ;  and 
in  like  manner  success  in  the  development 
of  mathematical  physics  will  depend 
equally  in  the  future  on  mathematical 
ability  of  the  highest  order.  In  this  line 
of  work  we  Americans  have  not  done  our 
full  duty,  and  it  behooves  us  as  mathe- 
maticians and  engineers,  now  that  we  have 
got  together  on  the  plane  of  mutual  in- 
terest, to  give  attention  to  this  important 
field  of  work. 

The  French  engineers  led  by  Navier  and 
followed  by  Lame,  Clapyron,  and  espe- 
cially by  the  "dean  of  elasticians, ' '  Barre 
de  Saint- Venant,  have  contributed  to  sci- 
ence the  most  important  branch  of  mathe- 
matical physics,  namely,  what  is  commonly 
called  the  theory  of  elasticity.  This  is 
superbly  difficult  in  its  purely  mathe- 
matical aspects  and  exquisitely  beautiful  in 
its  physical  aspects,  and  it  stands  as  a 
splendid  example  of  the  possibilities  which 
may  result  from  adequate  cooperation  be- 
tween mathematicians  and  engineers. 

The  chief  difficulty  in  the  way  of  de- 
veloping mathematical  physicists  appears 
to  lie  in  the  inadequate  appreciation  of 
this  type  of  work  by  contemporary  society. 
Pure  mathematics  has  a  prestige  of  more 
than  twenty  centuries  behind  it,  and 
the  practical  work  of  the  engineer  appeals 
even  to  the  dullest  of  intellects;  but  we 
have  failed  thus  far,  in  this  country  espe- 
cially, to  adequately  esteem  the  worker  in 
the  intermediate  field.  We  must  look  to 
it  that  more  attention  is  given  to  this  field 
in  our  colleges  and  universities.  Every 
university  should  have  two  or  three  men 
eminent  in  mathematical  physics  as  well 
as    two    or   three   men    eminent   in    pure 


mathematics.  Thus,  while  I  would  not 
advocate  the  pursuit  of  pure  mathematics 
or  the  pursuit  of  practical  engineering  less, 
I  would  urge  the  pursuit  of  mathematical 
physics  more.  It  is  only  by  the  cultiva- 
tion of  this  branch  of  study  and  investiga- 
tion that  we  can  keep  alive  the  sources  of 
engineering  knowledge.  Important  and 
indispensable  as  the  practical  work  of  the 
engineer  is,  the  cultivation  of  investigation 
and  discovery  in  his  science  is  still  more 
important  and  indispensable.  Hence  I 
would  urge  that  when  the  more  pressing 
questions  of  elementary  instruction  in 
mathematics  and  engineering  have  been 
adjusted,  we  give  attention  to  the  more 
inspiring  and  more  important  questions  of 
the  clarification  and  enlargement  of  the 
fundamental  ideas  of  our  sciences. 

R.  S.  Woodwakd 
Washington,  D.  C. 


THE     TEACHING     OF     MATHEMATICS     TO 
STUDENTS    OF    ENGINEERING1 

FROM    THE   STANDPOINT    OF   THE   PRACTISING 
ENGINEER 

I  am  honored  by  being  asked  to  say  a 
few  words  to  you  about  the  results  of  my 
experience  as  to  the  needs  of  the  teaching 
of  mathematics  to  students  of  engineering 
from  the  point  of  view  of  a  practical  engi- 
neer.    I  have  had  the  good  fortune  of  re- 

1  What  is  Needed  in  the  Teaching  of  Mathemat- 
ics to  Students  of  Engineering?  (a)  Range  of 
Subjects;  (b)  Extent  in  the  Various  Subjects; 
(c)  Methods  of  Presentation;  (d)  Chief  Aims. 
A  series  of  prepared  discussions  following  the 
formal  presentation  of  the  subject  by  Professor 
Edgar  J.  Townsend,  Professor  Alexander  Ziwet, 
Mr.  Charles  F.  Scott  and  President  Robert  S. 
Woodward.  (See  Science,  July  17,  1908,  pp.  69- 
79;  July  24,  1908,  pp.  109-113,  and  July  31,  1908, 
pp.  129-138.)  Presented  before  Sections  D  and  A 
of  the  American  Association  for  the  Advancement 
of  Science  and  the  Chicago  Section  of  the  American 
Mathematical  Society,  at  the  Chicago  meeting, 
December  31,  1907. 


SCIENCE 


27 


ceiving  quite  a  thorough  mathematical 
training  in  the  Ecole  des  Ponts  et 
Chaussees  of  France,  and  I  have  also  had 
the  good  fortune  of  developing  into  a  fairly- 
practical  engineer ;  my  remarks  will  there- 
fore be  backed  by  actual  experience. 

Mathematics  is  to  an  engineer  what 
anatomy  is  to  a  surgeon,  what  chemistry  is 
to  an  apothecary,  what  the  drill  is  to  an 
army  officer.  It  is  indispensable.  I  think 
we  all  agree  on  this  point. 

There  is  a  considerable  agitation  at  this 
time  in  France  and  Germany,  especially 
the  former,  favoring  the  limitation  of  the 
present  mathematical  program  of  the  engi- 
neering schools  on  the  ground  that  it  is 
unnecessarily  extensive.  From  personal 
observation,  I  can  say  that  the  program 
there  covers  a  considerably  wider  range 
than  in  the  average  American  college.  In 
the  first  place,  a  student  entering  an  engi- 
neering college  on  the  European  continent 
must  already  know  the  analytical  geom- 
etry, the  descriptive  geometry,  the  rudi- 
ments of  differential  and  integral  cal- 
culus, none  of  which  are  taught  here  until 
the  student  enters  college.  The  average 
length  of  a  college  engineering  course 
abroad  is  four  years,  one  of  the  exceptions 
being  the  Ecole  Centrale,  of  Paris,  France, 
where  the  course  is  only  three  years,  but 
where  the  entering  examinations  are  of  a 
comparatively  high  standard  and  the  stu- 
dents must  be  above  the  average  in  ability 
and  application  in  order  to  hold  their  own 
during  the  college  course.  It  is  obvious, 
therefore,  that  in  American  colleges,  time 
is  spent  on  pure  mathematics  which  could 
be  devoted  to  practical  study.  I  believe 
the  time  will  come  when  only  applied 
mathematics  will  be  taught  in  colleges,  and 
all  necessary  abstract  mathematics  will 
form  a  part  of  the  conditions  for  enter- 
ing. 

As  time  goes  on,  every  profession  tends 
more    and    more    towards    specialization. 


This  tendency  is  quite  marked  in  the  engi- 
neering profession.  It  would  take  too 
long  to  enumerate  all  of  these  special 
branches  of  engineering,  but  nearly  every 
branch  demands  a  somewhat  different 
mathematical  training.  The  time  may 
come  when  this  specialization  will  extend 
over  the  study  of  abstract  mathematics, 
differing  with  each  student  according  to 
the  branch  of  engineering  he  intends  to 
follow.  For  instance,  a  railway  engineer 
who  may  aspire  to  become  a  railroad 
official  requires  less  knowledge  of  calculus 
than  an  electrical  or  a  bridge  engineer;  on 
the  other  hand,  he  requires  a  greater 
knowledge  of  geology  than  the  electrical 
engineer,  and  a  greater  knowledge  of  com- 
mon law  than  the  bridge  engineer.  As 
my  remarks  are  merely  intended  to  fur- 
nish topics  for  discussion,  I  will  put  the 
following  question:  In  view  of  the  fact 
of  the  steadily  growing  scope  of  special 
education  will  it  be  desirable  and  possible 
to  specialize  mathematical  courses  in  col- 
leges and  adapt  them  to  each  branch  of 
engineering?  This,  as  I  understand,  is 
done  at  present  only  to  a  small  extent  in 
applied  mathematics. 

Bridge  engineering,  of  which  I  have 
made  a  specialty,  requires  probably  as  high 
a  mathematical  training  as  any  other 
branch  of  the  profession,  and  yet,  I  find 
that  part  of  the  higher  mathematics  which 
I  have  studied  in  college,  apart  from  the 
drilling  features  of  such  studies,  has  been 
entirely  useless ;  for  instance,  the  theory  of 
differential  equations.  The  time  I  spent 
on  it,  though  considerable,  was  not  suffi- 
cient to  make  me  understand  it  thoroughly, 
and  would  have  been  better  employed  in 
the  study  of  the  methods  of  least  work,  for 
instance,  which  no  bridge  engineer  should 
neglect  to  study. 

On  perusing  the  elementary  books  used 
in  high  schools,  I  have  been  often  struck 
with  the  dry,  uninteresting  manner  in  which 


28 


SCIENCE 


the  various  subjects  are  being  treated.  The 
examples  are  mostly  abstract,  very  few 
practical  problems  to  work  out.  Unless 
the  student  is  very  intelligent,  his  mind  re- 
tains nothing  beyond  a  chaos  of  formulae 
hard  to  remember  and  a  few  mechanical 
means  of  solving  abstract  problems.  He  is 
incapable  of  applying  an  equation  to  a 
practical  problem.  The  methods  of  pre- 
sentation should,  therefore,  be  such  that  the 
student  knows  the  why  and  wherefore  of 
each  operation — in  other  words,  that  he 
learns  to  think  mathematically.  This 
training  in  mathematical  thinking  should 
also  be  the  chief  aim:  one  does  not  know 
a  foreign  language  unless  one  is  able  to 
think  in  that  language ;  one  does  not  know, 
mathematics  unless  one  is  able  to  think 
mathematically.  It  is  not  necessary  for 
that  to  go  up  into  the  highest  mathematics, 
but  it  is  necessary  to  be  thoroughly  drilled 
in  elementary  principles  of  each  subject. 
These  elementary  principles  should  be- 
come a  second  nature  to  the  student,  just 
as  a  language  becomes  a  second  nature 
when  it  is  thoroughly  acquired.  Problems 
arise  every  day  in  the  practise  of  an  engi- 
neer, which  a  mathematical  mind  can  solve 
without  going  into  calculations,  such  prin- 
ciples as  those  of  maxima  and  minima, 
those  of  least  work,  of  cumulative  effect  of 
forces  and  others  are  invaluable  in  assist- 
ing to  arrive  at  a  logical  solution  of  many 
problems  without  the  use  of  a  scrap  of 
paper;  but  in  order  that  they  may  be  ap- 
plied, one  has  to  be  able  to  think  mathe- 
matically. With  a  proper  foundation,  the 
engineer's  mind  becomes  so  trained  that  he 
applies  those  fundamental  principles  un- 
consciously ;  they  direct  his  line  of  thought 
automatically,  so  to  speak.  How  to  secure 
such  a  foundation  in  a  student  must  be  left 
to  those  who  make  a  life-study  of  teach- 
ing. 

Ralph  Modjeski 
Chicago,  III. 


The  methods  of  teaching  mathematics  to 
engineering  students  in  vogue  twenty  years 
or  more  ago,  while  often  sufficiently 
strenuous,  were  invariably  far  from  satis- 
factory, in  that  they  failed  to  show  the  ap- 
plication of  the  subjects  to  engineering 
practise  and  to  explain  that  mathematical 
quantities  represent  something  real  and 
tangible,  not  merely  abstractions.  Possibly 
methods  have  changed  of  late  years;  but 
nothing  that  the  writer  has  seen  or  heard 
indicates  to  him  that  any  fundamental  im- 
provement has  been  effected.  Most  people 
continue  to  believe  that  mathematical  sub- 
jects are  taught  mainly  for  the  purpose  of 
training  the  mind,  and  that  the  manipula- 
tions involved  in  this  branch  of  science  are 
simply  mental  gymnastics.  Moreover,  even 
among  engineers  and  professors,  only  a  few 
recognize  adequately  the  great  importance 
of  mathematics  in  engineering  and  that  it 
is  something  real  and  substantial  instead  of 
fictitious  and  imaginary.  It  is  true  that 
higher  powers  than  the  third  are  not  con- 
ceivable entities;  but  the  mathematician 
recognizes  them  as  temporary  multiples  for 
future  reduction  to  entities. 

The  engineering  student  in  his  pure- 
mathematical  classes  is  not  taught  what 
equations  really  mean,  nor  what  are  their 
denominations  or  those  of  their  component 
parts.  All  that  he  learns  is  how  to  juggle 
with  quantities  in  order  to  produce  certain 
results.  It  is  left  to  the  professor  of 
rational  mechanics  to  teach  engineering 
students  the  reality  of  mathematics;  and 
too  often  he  fails  to  do  so,  sometimes,  per- 
haps, because  his  own  conception  thereof  is 
rather  vague. 

Concerning  the  teaching  of  pure  mathe- 
matics by  the  professor  of  rational  me- 
chanics the  writer  speaks  from  personal  ex- 
perience ;  for  more  than  a  quarter  of  a  cen- 
tury ago  he  taught  that  branch  of  engi- 
neering education  in  one  of  America 's  lead- 
ing   technical    schools.     Notwithstanding 


SCIENCE 


29 


the  fact  that  the  courses  in  pure  mathe- 
matics then  given  there  were  rigid  and  even 
severe,  the  students,  as  a  rule,  had  no  idea 
of  how  properly  to  apply  the  knowledge 
they  had  accumulated;  nor  did  they  know 
what  the  mathematical  terms  employed 
really  meant.  It  was  necessary  for  the 
writer  not  only  to  teach  his  own  branch, 
but  also  to  supplement  the  students '  knowl- 
edge of  pure  mathematics  by  explaining 
such  things  as  limits,  differential  coeffi- 
cients, total  and  partial  differentials,  and 
maxima  and  minima. 

Throughout  the  entire  course  in  rational 
mechanics  the  writer  either  demanded  from 
the  students  or  gave  them  demonstrations 
of  all  difficult  or  important  formulse;  and 
the  students  in  explaining  their  blackboard 
work  were  repeatedly  asked  to  state  the 
denominations,  not  only  of  the  equations  as 
a  whole,  but  also  of  their  factors  and  com- 
ponent parts.  The  answers  to  such  ques- 
tions evidenced  clearly  whether  the  student 
had  a  true  conception  of  the  mathematical 
work  he  was  doing,  or  whether  he  had 
merely  memorized  certain  manipulations  of 
quantities. 

It  was  the  writer's  custom  also  to  supple- 
ment as  much  as  possible  all  analytical 
work  by  graphical  demonstrations;  and  if 
he  were  to  resume  the  teaching  of  me- 
chanics, he  would  adhere  to  this  method. 

In  teaching  technical  mechanics  the 
writer  followed  only  to  a  certain  extent  the 
manner  of  instruction  just  described;  for 
by  the  time  his  students  had  reached  the 
technical  studies,  they  were  so  well  drilled 
and  weeded  out  that  constant  quizzing  on 
fundamentals  was  no  longer  necessary; 
nevertheless  the  question,  "what  is  the 
denomination  of  that  equation  or  of  that 
quantity,"  was  one  that  was  very  likely  to 
be  asked  any  student  who  gave  his  demon- 
strations haltingly  or  who  evidenced  at  all 
a  lack  of  conception  of  the  principles  in- 
volved. 


In  the  writer's  opinion,  the  manner  of 
teaching  pure  mathematics  to  engineering 
students  should  differ  materially  from  that 
usually  employed  in  academic  courses;  for 
while  in  the  latter  case  it  suffices  if  the 
instructors  be  good  mathematicians,  in  the 
former  they  should  also  be  engineers,  and 
should  have  taught,  or  at  least  should  have 
studied  specially,  both  rational  and  tech- 
nical mechanics. 

Some  institutions  still  adhere  to  the  anti- 
quated custom  of  teaching  pure  mathe- 
matics by  lectures.  This  method  has 
always  appeared  to  the  writer  to  be  per- 
fectly absurd;  for  the  primary  benefit  to 
be  obtained  from  the  study  of  mathematics 
is  mental  training ;  and  the  student  can  get 
this  only  by  severe  effort,  and  not  by  hav- 
ing another  man's  mind  do  the  reasoning 
for  him.  Midnight  oil  and  the  damp  towel 
are  for  most  students  necessary  accessories 
to  the  courses  in  pure  mathematics. 

The  writer  believes  that  the  only  legiti- 
mate lectures  in  pure-mathematical  courses 
for  engineering  students  are  as  follows : 

First;  A  short  opening  lecture  to  outline 
the  work  that  is  to  be  covered  in  the  course 
and  to  explain  how  best  to  study  the  sub- 
ject. 

Second:  Frequent  informal  talks  to  in- 
dicate the  application  of  the  mathematics 
studied  to  engineering  practise,  to  explain 
clearly  the  meaning  of  all  equations, 
factors  and  terms,  and  to  show  the  true 
raison  d'etre  of  all  that  is  being  done. 

Third:  A  concluding  lecture  in  the  na- 
ture of  a  resume  to  call  attention  to  what 
has  been  accomplished  during  the  entire 
course  and  to  the  importance  thereof. 

Fourth :  Personal  and  forcible  lectures  to 
lazy  students  so  as  to  give  them  clearly  to 
understand  that  they  must  either  study 
harder  or  drop  out  of  the  class. 

All  mathematical  work  done  by  engineer- 
ing students  should  be  so  thorough  and 
complete  that  the  subject  shall  be  almost  as 


so 


SCIENCE 


much  at  command  as  the  English  language 
or  the  four  simple  rules  of  arithmetic. 
Only  such  thorough  knowledge  will  enable 
the  engineer  to  use  mathematics  readily  as 
a  tool,  rather  than  as  a  final  resource  to  be 
employed  solely  in  extreme  need. 

Analytical  geometry  should  be  taught 
graphically  as  well  as  analytically  in  order 
that  the  student  shall  comprehend  it  fully 
and  shall  realize  that  the  work  is  real  and 
tangible  and  that  the  equations  represent 
lines,  surfaces,  and  volumes,  and  are  not  the 
results  of  mere  gymnastics.  A  knowledge 
of  the  graphics  of  analytical  geometry  is 
especially  valuable  in  mechanical  work,  in 
the  investigation  of  earth  pressures,  in 
suspension,  bridge  work,  and  in  many 
other  lines  of  engineering. 

The  proper  conception  of  the  meaning  of 
the  calculus  is  rarely  carried  away  by  the 
student.  He  knows  the  rules  and  can  per- 
form the  operations,  but  their  significance 
is  beyond  him;  consequently  he  does  halt- 
ingly and  bunglingly  the  original  work 
which  facility  in  the  use  of  the  calculus 
should  enable  him  to  perform  easily  and 
well.  This  state  of  affairs  is  a  crying  evil 
which  should  be  corrected  in  all  schools  that 
aim  to  give  first  class  engineering  courses. 

Descriptive  geometry  is  of  very  large 
value  in  the  preparation  of  drawings;  but, 
in  addition,  a  thorough  knowledge  of  it 
greatly  aids  in  the  conception  of  an  object 
in  space,  and,  consequently,  is  of  large 
assistance  in  the  evolution  of  original  de- 
signs. A  knowledge  of  it  prior  to  the 
study  of  the  courses  in  pure  mathematics 
assists  materially  in  the  conception  of  what 
the  latter  really  mean;  consequently  de- 
scriptive geometry  should  be  one  of  the 
earliest  courses  in  an  engineering  curric- 
ulum. 

A  sound  knowledge  of  mechanics,  the 
foundation  of  engineering,  is  impossible 
without  a  thorough  understanding  of 
mathematics.    It  is  true   that   mechanics 


may  be  learned  by  rote  or  by  so-called 
common-sense  methods;  but  the  "rule  of 
thumb"  or  "pocket-book"  engineer  never 
rises  to  noticeable  heights.  Such  an  engi- 
neer almost  invariably  fails  at  the  critical 
moment,  when  a  decision  must  be  sup- 
ported by  fundamental  principles.  It  is 
true  that  the  actual  use  of  analytical 
geometry,  calculus,  least  squares,  or  even 
higher  algebra  and  spherical  trigonometry, 
is  rare  in  the  practise  of  most  engineers; 
but  an  engineer's  grasp  of  technical  work 
depends  upon  his  knowledge  of  these  sub- 
jects; and  it  is  generally  conceded  that  a 
heavy  structure  can  not  be  continuously 
supported  on  a  weak  foundation. 

Mathematics  higher  than  the  calculus  is 
of  small  value  to  the  engineer,  except  pos- 
sibly as  a  training  for  the  mind;  but  the 
writer  is  of  the  opinion  that  any  such 
further  study  of  mathematics  is  a  detri- 
ment rather  than  a  help,  in  that  it  tends  to 
a  desire  to  reduce  all  work  to  mathematical 
calculation  and  thus  to  weaken  the  judg- 
ment. In  other  words,  excess  of  mathe- 
matical development  sometimes  produces 
an  unpractical  engineer. 

Most  graduate  engineers  immediately 
after  leaving  their  alma  mater  drop  for- 
ever the  study  of  mathematics,  both  pure 
and  applied,  except  in  so  far  as  they  are 
forced  to  use  them  by  their  professional 
work.  No  greater  mistake  than  this  can  be 
made,  for  it  takes  very  few  years  of  non- 
use  of  these  subjects  to  cause  one  to  forget 
them  utterly.  Every  young  engineer 
should  make  it  a  point  to  devote  a  certain 
portion  of  his  time  to  the  reviewing  of  the 
mathematical  studies  of  his  technical  course 
so  as  never  to  become  rusty  in  them;  and 
the  writer  believes  that  it  is  the  duty  of 
every  professor  of  mathematics  and  me- 
chanics to  impress  this  fact  continually 
upon  the  minds  of  his  students,  even  up  to 
the  very  day  of  their  graduation. 

Kansas  City,  Mo.        J-  A"  L'  Waddell 


SCIENCE 


31 


FROM    THE    STANDPOINT    OF    THE    PROFESSOR 
OF  ENGINEERING 

When  I  come  to  think  of  what  the  Math- 
ematical Society  has  brought  upon  itself, 
I  fear  that  it  may  feel  something  like  the 
football  when  it  is  kicked  back  and  forth 
upon  the  field.  On  the  one  hand  we  have 
the  trade-school  element  demanding  more 
knowledge  of  rules  and,  on  the  other,  the 
engineer  demanding  more  knowledge  of 
principles.  No  fair  discussion  of  this  sub- 
ject can  be  had  without  considering  for  a 
moment  the  conditions  and  definition  of 
engineering  itself.  The  most  common  defi- 
nition was  promulgated  more  than  half  a 
century  ago  by  Thomas  Tredgold,  to  the 
effect  that  civil  engineering,  which  was  the 
only  branch  of  engineering  then  known,  so 
the"  definition  may  be  considered  as  being 
general,  that  "civil  engineering  is  the  art 
of  directing  the  great  sources  of  power  in 
nature  to  the  use  and  convenience  of  man. ' ' 
I  should  say  that  "civil  engineering  to- 
day is  the  art  and  science  of  directing  the 
great  sources  of  power  in  nature  to  the  use 
and  convenience  of  man,"  and  from  that 
standpoint  I  am  willing  to  discuss  the  ques- 
tion as  to  how  much  and  how  far  mathe- 
matical instruction  should  enter. 

If  engineering  is  merely  an  art,  then 
mathematics  as  a  science  has  no  place  in 
the  training  of  the  engineer,  but  if  engi- 
neering is  a  science,  then  mathematics  has 
a  place.  Engineering  stands  to-day  in  the 
act  of  rising  to  the  status  of  a  science,  but 
is  still  hampered  by  the  tradesman.  On 
the  one  hand,  we  have  the  demand  that  the 
student's  training  be  such  as  primarily  to 
make  him  useful  to  some  one  to-morrow; 
and,  on  the  other  side,  that  it  make  him 
useful  to  the  world  perhaps  ten  years 
hence.  The  two  requirements  are  incon- 
sistent and  do  not  belong  together.  One  is 
that  of  the  trade  school,  and  many  should 
not  go  farther  than  that  because  they  have 
not  the  mental  capacity,  and  the  other  is 


the  demand  of  the  profession  into  which  a 
smaller  number  are  qualified  to  enter.  The 
trade  school  has  caused  most  of  the  trouble 
with  the  teaching  of  mathematics  because 
those  who  are  products  of  the  trade  school 
have  no  use  for  mathematics  as  a  science. 
The  complaint  about  the  teaching  of  mathe- 
matics does  not  come  from  engineers ;  they 
are  ready  to  use  mathematics  as  a  science. 
In  civil  engineering  it  is  fortunate  that  the 
profession  has  developed  along  lines  laid 
down  by  Rankine  rather  than  by  Traut- 
wine.  Both  have  had  their  use,  but  one 
of  them  produced  the  scientist  and  the 
other  produced  the  tradesman. 

It  is  maintained  in  the  institution  which 
I  have  the  honor  to  represent  that  they 
who  would  teach  engineering  must  prac- 
tise it,  and  by  analogy  we  might  say  that 
those  who  teach  mathematics  to  engineers 
should  themselves  be  engineers.  It  seems 
to  me  that  a  time  may  come  when  such  a 
condition  will  be  desirable,  but  let  me  say 
now  that  there  are  few  engineers  to-day 
who  have  had  sufficient  training  in  mathe- 
matics to  teach  it  themselves,  much  less  to 
tell  mathematicians  how  it  should  be 
taught.  We  can  perhaps  judge  of  the 
deficiency  of  the  student  who  comes  to  us, 
but  my  feeling  is  that  the  remedy  is  not  a 
question  of  what,  but  of  how.  Men  in  my 
institution  are  sending  us  students  well 
prepared  in  mathematics.  Others  do  not 
seem  to  be  so  fortunate.  Both  are  teach- 
ing the  same  subjects.  We  have  to  realize 
that  the  student  himself  is  a  factor  in  this 
question.  Some  students  become  mathe- 
maticians under  any  one;  others  would  not 
under  any  one.  To  be  taught  mathematics 
properly,  the  point  at  which  engineering 
minds  must  begin,  is  a  long  way  back.  I 
am  inclined  to  think  they  must  begin  some 
generations  before  birth.  The  mathematics 
of  grammar  schools  needs  overhauling  more 
than  the  mathematics  of  any  other  part  of 
our  educational  system,  and  probably  the 


32 


SCIENCE 


mathematics  of  high  schools  stands  next. 
The  essential  thing  that  we  ask  of  mathe- 
matics is  that  it  should  develop  the  quanti- 
tative reasoning  power,  and  the  student 
must  be  able  to  think  mathematically.  If 
he  has  not  acquired  that,  then  he  should 
drop  out  of  engineering  and  take  up  a 
trade.  It  was  mentioned  by  a  previous 
speaker  that  a  relatively  small  percentage 
of  the  graduates  from  a  certain  engineer- 
ing school  were  engaged  in  occupations  in 
which  mathematics  was  of  importance. 
From  a  somewhat  intimate  acquaintance 
with  the  graduates  of  that  institution,  I 
may  add  that  a  much  less  proportion  had 
sufficient  mathematical  training  to  take 
positions  in  which  mathematics  was  an  im- 
portant requirement.  Until  recently,  that 
college  has  stood  for  hardly  more  than  a 
highly  developed  trade  school,  and  it  is  not 
fair  to  cite  its  statistics  as  showing  condi- 
tions of  engineering  schools.  The  director 
of  that  institution  stated  many  years  ago 
that  he  did  not  consider  descriptive  geom- 
etry necessary  for  mechanical  engineers, 
and  his  students,  having  had  their  course 
in  machine  design  in  the  junior  year  were 
frequently  found  taking  their  only  course 
of  descriptive  geometry  when  seniors. 

The  question  has  been  raised  as  to  the 
increase  of  mathematics  for  entrance  to 
engineering  schools.  My  view  of  that  is 
that  it  would  not  be  wise  to  raise  the 
requirements  at  this  time.  Cornell  has,  it 
is  true,  increased  the  requirements,  but  at 
the  sacrifice  of  both  physics  and  chemistry, 
and  to  my  mind  it  is  best  that  physics  and 
chemistry  be  taught  at  the  age  of  high 
school  students,  rather  than  analytics  and 
trigonometry.  If  you  can  not  do  both  it 
is  better  that  the  young  mind  have  im- 
pressed upon  it  some  physical  science 
rather  than  encounter  the  more  abstract 
demands  of  mathematics.  In  the  training 
of  students  in  mathematics  I  would  wipe 
out  formulae.     We  want  principles.    There 


is  generally  taught  too  much  of  the  for- 
mula, as  that  is  what  the  trade  school  has 
demanded.  Some  have  objected  to  the 
statement  that  mathematics  should  be  a 
tool.  To  my  mind  it  is  certainly  an  in- 
strument. It  is  one  of  the  things  that  the 
engineer  must  use,  and  in  order  that  he 
may  use  it,  he  must  be  sufficiently  familiar 
with  it,  so  that  it  will  respond  to  his  use 
when  he  desires  it.  The  question  of  elec- 
tion in  mathematics  has  been  suggested. 
I  am  certainly  favorable  to  elections  in  that 
subject,  but  I  question  the  advisability  of 
such  opportunity  in  any  subject  for  the 
ordinary  student,  before  the  fourth  year. 
My  own  observation  leads  me  to  conclude 
that  very  few  students  are  able  to  elect 
intelligently  before  that  time.  The  re- 
marks relative  to  the  employment  of  inex- 
perienced instructors  instead  of  competent 
professors  show  a  fault  to  lie  with  the 
heads  of  the  various  departments  them- 
selves. If  they  are  willing  to  accept,  for 
the  purpose  of  instructing  students,  the 
men  who  have  been  unable  to  find  positions 
elsewhere,  and  employ  only  such  as  will 
work  for  seven  to  nine  hundred  dollars  per 
year,  the  unsatisfactory  results  are  their 
own  fault.  The  responsible  parties,  the 
trustees  and  regents  of  educational  institu- 
tions, will  furnish  what  is  shown  to  be 
necessary.  If  it  is  necessary  that  you  have 
better  men,  then  say  so  and  get  them,  but 
if  you  are  satisfied  with  what  you  now 
have,  then  you  can  expect  to  see  decorative 
cornices  and  stained  glass  windows,  rather 
than  intellect  and  culture,  the  characteris- 
tics of  our  universities. 

Gardner  S.  Williams 
Univeesity  of  Michigan 


It  may  save  time  to  state  briefly  at  the 
beginning  my  thought  on  what  is  needed 
in  the  teaching  of  mathematics  to  engi- 
neering students.      It  seems  to  me  that, 


SCIENCE 


33 


outside  of  the  general  cultural  and  devel- 
opmental purpose  of  the  study  of  mathe- 
matics, the  instruction  of  engineering  stu- 
dents may  be  discussed  under  three  dif- 
ferent phases,  which  for  want  of  better 
terma^may  be  named:  (1)  theory,  (2) 
practise,  (3)  philosophy;  that  successful 
teaching  of  mathematics  to  engineering 
students  depends  upon  giving  the  right 
relative  proportion  or  emphasis  to  these 
three  phases  of  instruction;  that  the  con- 
tent of  the  instruction,  within  the  limits 
of  present  usage  in  engineering  schools,  is 
of  minor  importance;  that  thoroughness  is 
essential,  and  that  it  is  better  to  cut  down 
the  extent  of  the  matter  gone  over  if  there- 
by a  more  thorough  grasp  of  the  subject 
is  secured;  and  that  the  instructor  must 
always  keep  in  mind  that  he.  is  training  an 
average  boy  of  average  preparation  with 
a  view  to  using  mathematical  principles 
and  methods  of  attack  and  mathematical 
operations  and  conceptions  in  the  mastery 
of  his  engineering  studies  and  in  the  treat- 
ment of  the  varied  problems  which  will 
arise  in  his  later  engineering  experience. 

The  great  mass  of  our  engineering  stu- 
dents, like  the  great  mass  of  our  engineers, 
are  not  mathematical  geniuses.  In  the  dis- 
cussion of  the  subject  we  must  keep  ever 
in  mind  that  the  average  engineering  stu- 
dent is  not  of  strong  mathematical  bent. 
Many  of  those  with  only  mediocre  mathe- 
matical ability  make  successful  engineers, 
and  the  student  of  strong  mathematical 
turn  may  lack  in  some  direction  or  may 
have  a  disproportionate  measure  of  the 
importance  of  his  analytical  powers  and 
drop  behind  his  less  mathematical  class- 
mate. I  want  to  make  a  plea  for  the  aver- 
age student;  the  boy  whose  analytical 
powers  have  to  be  encouraged  and  devel- 
oped. The  methods  of  presentation  must 
be  made  elastic  enough  to  include  this  great 
class  of  students,  or  we  shall  fail  to  do  our 
duty  as  teachers. 


I  have  mentioned  three  phases  in  the 
presentation  of  mathematical  subjects. 
These  may  be  considered  in  order.  It  must 
be  understood  that  these  phases  are  not 
mutually  exclusive. 

1.  Theory.— Analysis,  demonstration  and 
the  general  derivation  and  presentation  of 
mathematical  principles.  The  derivation 
and  exposition  of  mathematical  principles 
and  operations  and  the  appreciation  of 
mathematical  concepts  are  universally 
accepted  as  important  elements  in  the 
education  of  an  engineer.  The  use  of 
mathematical  forms  of  attack,  the  training 
in  processes  of  reasoning,  the  formation 
of  logical  habits  of  thought,  are  hardly  sec- 
ondary in  importance.  And  yet  much  less 
emphasis  is  placed  on  formal  demonstra- 
tion and  reasoning  than  formerly — fre- 
quently this  element  is  overlooked  or 
treated  in  a  slipshod  way.  The  student 
comes  to  feel  that  he  is  after  facts  and  that 
the  derivation  and  proof  of  principles  in- 
volves useless  effort— he  is  willing  to  accept 
their  authenticity.  It  may  be  that  years 
ago  our  instructional  methods  carried  for- 
mal processes  to  an  extreme  and  that  as  a 
result  mathematical  work  became  meaning- 
less lingo  or  memorized  facts  to  many  stu- 
dents. This  does  not  furnish  argument 
for  the  abandonment  of  training  in  formal 
reasoning.  For  the  young  mind,  practise 
in  analysis,  in  formal  demonstration  is 
illuminating  and  developing.  Even  the 
repetitive  forms  of  analysis  in  the  old-time 
mental  arithmetic  had  great  mathematical 
educational  value.  The  speaker  feels  that 
in  the  effort  to  avoid  barren  formalism  the 
pendulum  has  swung  too  far  the  other  way, 
and  that  both  in  high  school  and  in  tech- 
nical school,  and  in  the  applied  engineer- 
ing subjects  as  well,  the  training  in  an- 
alytical methods  and  formal  processes  is 
weak.  He  believes  that  good  results  would 
follow  putting   greater   emphasis   on  this 


34 


SCIENCE 


phase  of  instruction  than  now  seems  to  be 
the  trend. 

2.  Practise.— The  use  and  applicability 
of  mathematical  principles  and  processes 
in  the  solution  of  problems,  drill  on  these 
principles,  and  the  acquisition  of  facility 
in  their  use.  To  the  average  student  the 
working  of  examples  is  illuminating. 
Without  it  the  concept  is  but  vaguely  com- 
prehended, the  derivation  only  faintly 
understood,  the  process  may  seem  merely 
verbal  legerdemain.  Properly  used,  this 
phase  of  mathematical  instruction  is  of 
great  advantage  to  the  student  of  average 
mathematical  ability.  It  opens  up  the 
view;  it  clears  away  uncertainties;  it  fixes 
principles  and  concepts ;  it  gives  life  to  the 
subject.  The  problems  used  should  be 
within  the  field  of  the  students'  experience 
and  comprehension  and  may  well  bear  some 
relation  to  his  future  work,  both  in  the 
engineering  class-room  and  beyond.  And 
the  second  part  of  this  heading  is  not  less 
important.  Mathematics  is  a  tool  for  the 
engineering  student,  and  he  must  acquire 
facility  in  its  use.  This  does  not  mean 
that  the  instructor  should  attempt  to  make 
him  a  finished  calculator  or  an  expert 
workman — time  is  too  short— but  mathe- 
matical principles  and  processes  must  be 
more  to  the  student  than  a  vague  some- 
thing which  he  recognizes  when  his  atten- 
tion is  directed  thereto.  Instead,  he  must 
have  a  mastery  of  at  least  the  fundamen- 
tals and  he  must  be  able  to  use  such  prin- 
ciples and  processes  in  his  later  studies 
without  having  to  divert  his  attention  and 
energy  too  much  from  the  engineering  fea- 
tures involved.  To  acquire  this  facility 
requires  drill  and  repetition,  and  this  drill 
must  constitute  a  part  of  the  mathematical 
training  of  the  engineering  student.  The 
multiplication  table  had  to  be  learned,  and 
many  other  important  things  have  to  be 
acquired  in  the  same  way. 

But  it  seems  that  this  important  side  of 


instruction  may  be  abused.  The  student 
who  thinks  that  to  accept  facts  and  work 
problems  is  sufficient  and  the  instructor 
who  thinks  that  illustrations  and  practise 
work  alone  constitute  mathematical  train- 
ing or  that  mere  laboratory  methods  suffice 
are  greatly  mistaken.  The  mere  substitu- 
tion in  formulas  is  only  rule-of-thumb 
work,  so  much  decried  in  engineering;  and 
the  mechanic  who  knows  how  to  use  tools, 
and  no  more,  is  not  an  engineer.  There 
must  be  a  direct  connection  with  the  theory 
and  the  philosophy  of  the  subject  to  make 
the  practise  side  serve  its  proper  purpose. 
In  teaching  mathematics  years  ago,  expres- 
sions of  approval  came  to  me  because  I 
was  so  "practical,"  but  the  underlying 
purpose  of  the  practical  part  was  not 
always  understood,  though  this  lack  of 
understanding  did  not  affect  the  results  of 
the  method.  Inside  the  "sugar  coating' r 
there  should  always  be  a  principle  to  fix, 
a  concept  to  illumine,  a  process  to  ex- 
emplify, a  derivation  to  expound.  There 
seems  to  be  a  tendency  among  some  to  over- 
do this  side  of  the  work  to  the  detriment 
of  the  first  side.  While  the  practise  fea- 
ture is  a  valuable  auxiliary  in  mathemat- 
ical instruction,  it  should  never  be  the  lead- 
ing motive.  Student  and  instructor  alike 
should  recognize  this. 

3.  Philosophy  of  the  Subject. — The  basis 
on  which  the  science  rests,  the  underlying 
meaning  of  the  mathematical  processes 
used,  a  philosophical  study  of  the  method 
of  treatment  and  of  the  concepts  used, 
their  connection  with  related  things.  This 
is  difficult  to  discuss  in  a  general  way,  and 
of  course  this  phase  is  intimately  connected 
with  the  first  and  second.  To  my  mind 
this  phase  should  not  be  neglected.  It 
must  be  apportioned  according  to  the  abil- 
ity of  the  student.  An  understanding  of 
the  philosophy  of  the  subject  will  widen 
his  field  of  view  and  lessen  the  chances  of 
error.      The  better  grasp  of  the  meaning 


SCIENCE 


35 


will  be  advantageous.  Its  presentation 
involves  difficulties,  and  text-books  gener- 
ally disregard  it.  It  must  not  be  over- 
emphasized, as  is  illustrated  by  the  treat- 
ment in  a  recent  text-book  in  applied  math- 
ematics, where  it  is  used  largely  to  the  ex- 
clusion of  analysis  and  demonstration. 

Effective  methods  in  mathematical  sub- 
jects involve,  then,  the  skillful  selection  in 
proper  proportion  from  these  three  phases, 
and  the  best  teacher  will  make  for  himself 
the  best  selection.  The  derivation  and 
elucidation  of  mathematical  principles, 
facility  in  their  use  and  application,  and 
an  understanding  of  the  basis  on  which 
principles  and  methods  rest  are  all  essen- 
tial. A  good  text-book— one  properly  pro- 
portioned—aids greatly  in  the  work  of  in- 
struction. However,  it  is  the  teacher  on 
whom  reliance  is  placed  in  the  end,  and  for 
the  student  of  average  mathematical  abil- 
ity the  teacher's  influence  constitutes  a 
large  element.  It  is  highly  advantageous 
for  the  teacher  to  have  a  fair  knowledge 
of  the  applications  of  mathematics  which 
the  student  will  make  in  later  work  and 
to  have  sympathy  and  interest  in  such 
work.  Let  us  also  emphasize  the  impor- 
tance of  having  the  best  of  teachers  for 
mathematical  instruction. 

Let  me  add  to  this  that  it  is  my  belief, 
growing  stronger  after  many  years  of  ob- 
servation, that  the  average  engineering 
student  gets  relatively  little  from  lectures 
on  mathematical  subjects;  that  many  in- 
structors talk  too  much  themselves;  that 
the  student  must  have  the  opportunity  to 
express  himself  and  must  be  required  to 
use  the  mathematical  language  and  to  try 
his  own  skill,  and  this  in  other  than  formal 
quizzes ;  and  that  recitation  and  drill  work 
are  essential  factors  in  giving  training  to 
this  average  student. 

Little  can  be  said  in  the  time  at  my  dis- 
posal on  the  ground  which  should  be  cov- 
ered   in    mathematical    instruction.      Two 


classes  of  matter  are  studied:  (1)  funda- 
mental principles  forming  the  skeleton  of 
the  work,  and  (2)  the  more  complicated 
topics,  involving  further  detail  and  insight. 
There  will  be  little  difference  of  opinion 
on  the  first  class.  There  will  be  more  on 
the  second.  I  have  found  in  the  teaching 
of  mechanics  and  of  various  engineering 
subjects  that  certain  topics  and  methods 
not  ordinarily  given  in  mathematical  in- 
struction may  advantageously  be  used  in 
the  presentation  of  the  work.  The  teacher 
of  thermo-dynamics  or  of  electro-dynamics 
has  other  topics  to  suggest,  and  still  other 
topics  will  come  from  other  sources.  Not 
all  of  these  may  be  allowed.  In  fact,  it 
makes  little  difference  what  particular 
topics  are  included  so  long  as  the  student 
has  thorough  training  in  some  of  the  more 
complex  work.  The  difficulty  of  giving 
instruction  in  complex  work  lies  not  so 
much  in  the  time  required,  as  in  the  ob- 
stacle that  the  concepts  lie  beyond  the 
student's  experience  and  that  he  is  not 
ready  to  comprehend  their  meaning.  If 
he  had  the  opportunity  to  study  these  top- 
ics after  he  has  reached  the  subject  in 
which  they  are  to  be  used,  or  if  he  could 
go  back  over  a  part  of  mathematics  after 
his  study  has  taken  him  into  their  field  of 
application,  as  indeed  his  instructor  has 
done  for  himself,  the  result  would  be  more 
satisfactory.  All  these  limitations  must  be 
considered  in  choosing  the  ground  to  be 
covered  in  mathematical  instruction. 

Arthur  N.  Talbot 
Univebsity  of  Illinois 


THE    TEACHING    OF    MATHEMATICS    TO 
STUDENTS   OF   ENGINEERING1 

FROM    THE    STANDPOINT    OP    THE    PROFESSOR 
OF  ENGINEERING 

I  feel  that  in  this  discussion  we  engi- 
neers occupy  rather  an  unfortunate  posi- 
tion, on  account  of  the  fact  that  we  are 
1  Continued  from  the  issue  of  August  7. 


36 


SCIENCE 


compelled  to  assume  the  position  of  critics. 
The  student  comes  to  us  from  the  teachers 
of  mathematics,  presumably  equipped  with 
a  knowledge  of  that  subject,  and  it  becomes 
our  duty  to  teach  him  subjects  in  which  he 
makes  use  of  this  preparation,  and  to  find 
out  whether  he  has  learned  to  use  mathe- 
matics as  a  tool.  However,  I  believe  that 
only  by  friendly  criticism  can  progress  be 
made,  and  that  every  one  ought  to  be  will- 
ing to  accept  such  criticism  when  given  in 
the  proper  spirit.  I  had  much  rather  be 
criticized  than  criticize  others,  and  we 
teachers  of  engineering  hope  that  we  are 
always  ready  to  receive  suggestions,  not 
only  from  other  teachers,  but  from  prac- 
tising engineers. 

I  must  first  insist  that  for  the  engineer 
mathematics  is  to  be  regarded  as  a  tool — 
not  as  something  which  is  studied  simply 
for  the  development  of  some  mental  powers, 
but  for  the  ability  which  it  ought  to  give 
a  man  to  do  something— to  use  the  results 
and  methods  which  he  has  been  taught  in 
solving  the  problems  of  his  profession. 

There  has  been  a  good  deal  of  discussion 
in  the  past  as  to  the  value  of  mathematics 
simply  as  a  means  of  mental  training,  with- 
out reference  to  its  use,  and  perhaps  most 
of  us  remember  the  paper  by  Sir  William 
Hamilton  written  seventy-six  years  ago,  in 
which  he  maintains  that  there  is  no  one 
of  the  subjects  in  the  curriculum  which 
develops  a  smaller  number  of  mental  facul- 
ties or  develops  them  in  a  more  imperfect 
and  inadequate  manner  than  mathematics. 
I  have  never  seen  what  has  seemed  to  me  a 
conclusive  refutation  of  Sir  William 
Hamilton's  main  arguments,  and  for  my 
part  I  am  disposed  to  agree  with  him  in 
general,  and  to  assign  a  comparatively  low 
value  to  mathematics  simply  as  a  training, 
aside  from  its  applications.  I  have  not 
observed  that  students  trained  in  this  sub- 
ject are  able  to  reason  any  better  than  stu- 
dents who  have  ignored  mathematics;  in- 


deed, I  believe  that  many  non-mathe- 
matical subjects  afford  a  better  training  in 
reasoning  than  the  study  of  mathematics. 
This  view  may  perhaps  be  justified  by  re- 
membering that  mathematics,  aside  from 
geometry,  deals  with  questions  of  quantity 
and  number,  but  not  with  questions  of 
quality.  The  student  puts  certain  fixed 
data  into  his  mathematical  machine  and 
grinds  out  the  result.  He  does  not  learn 
to  observe  and  to  discover  the  finer  and 
more  elusive,  but  equally  important, 
sources  of  error  likely  to  occur  in  the  ordi- 
nary questions  of  daily  life,  because  he  is 
dealing  with  a  rigid,  unyielding,  logical 
machine.  In  this  way  his  mind  may  be- 
come hardened — he  deals  with  rigid  demon- 
strations and  is  unwilling  or  unable  to  ap- 
preciate or  submit  to  a  less  rigid  method, 
which  is  often  the  only  possible  one.  The 
best  student  of  mathematics  is  frequently 
one  of  the  poorest  of  engineers.  Give  him 
fixed  data  and  he  will  get  the  proper  result, 
but  he  may  be  entirely  incapable  of  attack- 
ing a  practical  problem,  or  of  deciding 
what  the  proper  data  are. 

I  have  not  observed  that  students  of 
mathematics  are,  as  a  rule,  more  accurate 
than  other  students,  or  that  a  training  in 
the  branches  of  mathematics  above  arith- 
metic leads  to  accuracy.  Indeed,  it  more 
often  appears  to  pervert  the  sense  of  per- 
spective, and  to  lead  students  to  work  out  a 
result  to  several  figures  in  cases  where  a 
smaller  number  only  may  be  significant. 
Mathematics  does  not  train  the  observation, 
neither  does  it  train  the  imagination,  except 
in  the  geometrical  branches,  which  are  now 
comparatively  neglected  since  the  powerful 
modern  methods  in  analysis  have  been  in- 
troduced. 

Hamilton  only  allowed,  as  I  remember, 
that  mathematics  adequately  trained  one 
faculty,  namely,  that  of  continuous  atten- 
tion: but  I  fail  to  see  that  this  is  trained 
any  better  by  the  study  of  mathematics 


SCIENCE 


37 


than  by  that  of  language,  chemistry  or  by 
other  natural  sciences.  Unfortunately,  as 
at  present  taught  it  does  train  the  memory, 
in  a  way  that  it  ought  not  to  do.  The 
ordinary  student  of  mathematics  subor- 
dinates perception  to  a  ynemorization  of 
formulae  and  rules. 

I  believe,  therefore,  that  from  the  point 
of  view  of  the  engineer,  mathematics  should 
be  taught  with  the  object  of  giving  the 
student  power  to  use  it  as  a  tool.  With 
reference  to  this  I  think  it  is  fair  to  say 
that  the  consensus  of  opinion  among  engi- 
neering teachers  and  practitioners  is  that 
the  results  of  the  present  mathematical 
training  are  very  poor.  The  average  stu- 
dent who  has  completed  his  mathematical 
course  is  frequently  quite  helpless  when 
called  upon  to  attack  a  concrete  engineer- 
ing problem,  and  it  is  a  common  remark 
by  civil  engineering  students  that  they  did 
not  really  learn  any  mathematics  until  they 
studied  mechanics  or  the  theory  of  struc- 
tures. The  results  seem  to  be  almost 
equally  poor  no  matter  what  institution  the 
student  comes  from,  for  in  my  classes  there 
have  been  students  from  most  of  the  prin- 
cipal universities  and  technical  schools  in 
the  country  and  I  have  failed  to  notice  any 
great  difference  in  them  in  that  respect. 
They  very  generally  lack  the  power  to 
do  anything  with  the  mathematics  which 
they  have  been  taught. 

With  reference  to  the  reasons  for  this 
state'  of  things,  I  venture  to  state  what 
seem  to  me  to  be  some  of  them,  and  the 
suggestions  which  have  occurred  to  me  by 
which  possibly  the  results  might  be  im- 
proved. 

1.  In  one  of  the  previous  papers  a  state- 
ment was  made  that  many  students  who 
studied  advanced  algebra  in  the  technical 
schools  had  not  studied  algebra  in  the  pre- 
paratory schools  for  the  two  years  previous. 
This  illustrates  what  I  believe  to  be  one 
failing  in  our  so-called  system  of  educa- 


tion, namely,  the  lack  of  continuity.  The 
remedy  is  to  reform  and  simplify  the  cur- 
riculum, and  to  unify  and  simplify  the 
entrance  examinations  to  our  colleges  and 
technical  schools.  So  long  as  these  en- 
trance examinations  are  so  extended  and 
cover  so  large  a  range  of  subjects,  our  pre- 
paratory schools  will  be  unable  to  carry 
out  their  true  purpose,  which  is,  as  it  seems 
to  me,  no  less  and  no  more  than  that  of 
all  education,  namely,  to  train  a  man 
thoroughly  in  a  few  things  and  to  give  him 
the  power  to  do  some  little  thinking  for 
himself  and  to  take  up  new  subjects  with- 
out assistance. 

2.  The  great  inherent  difficulty  which 
teachers  of  mathematics  as  well  as  teachers 
of  every  other  subject  meet  with  is  the 
attitude  of  the  student,  and  his  inability 
to  realize  the  seriousness  and  the  im- 
portance of  his  work.  I  am  fond  of  ex- 
pressing my  view  in  regard  to  this  by  the 
statement  that  the  school  is  not  a  restau- 
rant, but  a  gymnasium ;  not  a  place  where 
a  student  comes  to  be  filled  up,  but  a  place 
where  he  finds  apparatus  and  the  instruc- 
tion, by  making  use  of  which  he  may 
strengthen  his  mental  muscles. 

The  manufacturer  can  take  his  raw  ma- 
terial and  shape  it  into  the  form  which 
he  desires.  The  raw  material  of  the 
teacher  is  the  student,  but  the  teacher  can 
not  take  this  material  and  shape  it ;  he  can 
only  show  it  how  it  can  shape  itself.  I 
believe,  however,  that  much  may  be  done 
in  impressing  upon  students  the  proper 
attitude  which  they  should  take  toward 
their  work,  and  by  a  proper  cooperation 
between  teachers  and  parents,  which  is  un- 
fortunately lacking  as  a  rule  in  this 
country,  and  the  responsibility  for  which 
must  largely  fall  upon  the  parents. 

3.  I  believe  that  one  cause  of  the  poor 
results  in  mathematical  teaching  is  that  too 
great  a  stress  is  laid  upon  analysis.  Mathe- 
matics is,  of  course,  divided  into  geometry 


38 


SCIENCE 


and  calculus,  using  the  words  in  their 
widest  sense.  Geometry  is  concrete;  and 
the  mind  perceives  the  steps  in  a  geo- 
metrical demonstration.  This  branch,  the 
oldest  branch  of  mathematics,  however,  has 
been  largely  supplanted  by  the  modern 
analytic  methods  which  have  been  de- 
veloped during  the  past  three  centuries, 
largely  to  the  detriment,  it  seems  to  me,  of 
the  educational  results  obtained.  Analysis 
is  abstract— it  is  a  powerful  machine,  an 
invention  for  doing  certain  things.  Into 
one  end  of  the  machine  we  put  the  data; 
we  turn  the  crank,  and  the  result  comes 
out  with  absolute  correctness  so  far  as  is 
warranted  by  the  data.  Now  I  believe  that 
too  much  stress  is  laid  on  these  analytical 
processes;  that  the  student  is  not  urged  to 
visualize  his  results,  to  express  them 
geometrically  and  to  interpret  his  equa- 
tions. I  warmly  second  the  remarks  of 
Professor  Ziwet  with  reference  to  descrip- 
tive geometry,  which  I  believe  should  be 
treated  as  a  branch  of  mathematics  and 
taught  more  thoroughly,  as  it  is  taught  in 
Germany.  For  my  part,  I  derived  as  much 
benefit  from  my  study  of  descriptive 
geometry,  and  afterward  from  the  study 
of  projective  geometry,  as  from  any  other 
mathematical  studies.  These  studies  train 
the  imagination,  which  analysis  does  not 
do.  But  in  the  use  of  analysis,  the  first 
step,  namely,  the  formulation  of  a  problem, 
is  really  concrete.  This,  too,  is  neglected 
in  our  usual  courses.  Our  examination 
papers  are  full  of  questions  which  involve 
simply  the  analytic  processes— the  differ- 
entiation, the  integration,  the  twisting  and 
turning  of  equations,  while  much  less  at- 
tention is  paid  to  the  formulation  in  mathe- 
matical language  of  practical  problems. 
Our  students,  therefore,  when  they  meet  a 
practical  problem,  are  unable  to  select  or 
judge  of  the  correctness  of  the  data,  and 
even  if  they  can  do  this,  are  unable  to 
formulate  the  data  as  a  preliminary  to  the 


solving  of  the  problem  by  the  use  of  the 
mathematical  machine. 

One  of  the  great  defects  which  I  find  in 
students  of  mathematics  is  one  already  re- 
ferred to,  namely,  that  they  do  not  in- 
terpret their  equations.  The  average  stu- 
dent who  has  completed  his  mathematical 
course,  for  instance,  has  not  the  slightest 
conception  of  what  a  parabola  is.  I  make 
this  statement  advisedly,  because  I  have 
tested  it  again  and  again  for  years.  If  he 
could  tell  you  what  a  parabola  really  is  in 
his  mind,  he  would  probably  tell  you  that 
it  was  a  curve  of  more  or  less  beauty 
represented  by  letters.  Perhaps  he  could 
tell  you  what  the  letters  are,  but  give  him  a 
concrete  problem  and  he  would  convince 
you  immediately  that  he  did  not  know 
what  the  letters  mean. 

4.  Another  defect,  as  it  seems  to  me,  in 
our  present  methods,  is  the  lack  of  training 
in  mental  operations.  In  the  good  old 
days  menial  arithmetic  was  taught,  but 
that  seems  to  have  gone  out  of  fashion, 
with  so  many  of  the  other  good  old 
methods.  Ask  the  ordinary  graduate  of 
our  mathematical  courses  to  tell  you  the 
square  of  20.75  without  using  pencil  or 
paper  and  he  will  look  at  you  open- 
mouthed  with  astonishment,  but  if  he  had 
really  grasped  the  meaning  of  the  binomial 
theorem  and  had  learned  to  do  a  few 
"sums"  in  his  head,  any  grammar-school 
boy  would,  of  course,  be  able  to  give  the 
result  immediately. 

5.  Another  reason  for  poor  results  is,  I 
believe,  inadequate  class-room  methods,  and 
especially  the  use  of  the  lecture  system. 
In  Germany,  where  the  students  in  the  uni- 
versities have  had  the  advantage  of  a 
thorough  preliminary  training,  they  may 
be  able  to  appreciate  lectures  on  mathe- 
matical subjects,  although  I  doubt  even 
this  in  the  case  of  the  average  student. 
For  students  in  our  American  universities, 
however,  I  believe  that  lectures  in  mathe- 


SCIENCE 


39 


matics  are  almost  useless,  except  for  a  very 
small  number  of  students;  and  yet,  I  am 
told  that  even  in  some  of  our  high  schools 
mathematics  is  taught  to  a  considerable 
extent  by  lectures.  The  lecture  system  is 
easy  for  the  teacher.  It  involves  no  cross- 
questioning,  no  endeavor  to  discern  what 
is  going  on  in  the  student's  mind,  no  adap- 
tation of  question  with  the  object  of  put- 
ting him  on  the  right  track. 

Again,  some  mathematical  exercises  are 
conducted  by  sending  the  students  to  the 
board,  each  with  a  problem  to  solve,  and 
then  marking  that  on  the  correctness  of 
their  work.  Occasionally  a  formal  expla- 
nation of  his  problem  is  required  of  the 
student.  This,  again,  seems  to  me  to  be  a 
mistaken  method.  Many  a  student  can  go 
through  a  demonstration  of  a  principle,  or 
solve  a  problem  by  substitution  in  a 
formula,  while  knowing  nothing  of  the  real 
meaning  of  the  subject.  In  my  opinion 
class-room  instruction  should  be  conducted 
by  the  Socratic  method— by  question  and 
answer— the  teacher  endeavoring  to  put 
and  keep  the  student  upon  the  right  track 
by  showing  him  what  he  can  do  for  him- 
self if  he  will  only  learn  how. 

6.  Eeference  has  been  made  to  the  kind 
of  teachers  of  mathematics.  Personally  I 
believe  that  in  teaching  the  subject  to  engi- 
neering students  the  best  results  would  be 
obtained  if  the  teachers  were  engineers,  or 
at  least  if  they  were  near  enough  to  being 
engineers  to  take  an  interest  in  the  con- 
crete problems  themselves  as  distinct  from 
their  solution.  If  I  am  correct  in  the  be- 
lief that  mathematics  should  be  taught  as  a 
tool,  then  it  can  be  taught  best  by  those 
who  know  how  to  use  it  as  a  tool.  Un- 
fortunately, however,  it  is  difficult  to  get 
engineers  who  are  sufficiently  interested  in 
mathematics  and  sufficiently  masters  of 
that  subject,  who  are  willing  to  devote 
themselves  to  teaching.  The  men  who  are 
interested  in  the  problems  prefer  to  devote 


themselves  to  those  problems,  and  to  go 
into  practical  work.  It  is  not  necessary, 
however,  as  suggested  above,  that  the 
teachers  of  mathematics  should  be  engi- 
neers if  only  they  will  take  an  interest  in 
the  problems  themselves,  and  in  the  point 
of  view  which  the  student  should  take. 
They  can  do  this  by  cooperation  with  the 
engineering  teachers,  by  attending  engi- 
neering courses,  and,  perhaps,  by  a  little 
more  realization  than  they  now  have  that 
their  work  is  preliminary  to  other  and 
more  important  work,  and  that  as  a  matter 
of  fact  if  the  engineering  student  does  not 
learn  to  use  his  mathematics  as  a  tool  it 
is  practically  of  no  value  to  him.  For  the 
engineer,  mathematics  is  the  servant,  and 
the  mathematical  teacher  should  aim  to 
teach  the  subject  in  such  a  way  as  to  obtain 
as  nearly  as  possible  the  results  which  in- 
telligent engineering  teachers  and  practi- 
tioners desire  to  have  obtained. 

George  F.  Swain 
Massachusetts  Institute  of  Technology 

FROM  THE  STANDPOINT  OP  THE  PROFESSOR  OF 

MATHEMATICS  IN  THE  ENGINEERING 

COLLEGE 

We  must  not  take  too  seriously  what 
engineers  have  to  say  in  an  educational 
discussion,  nor  take  too  much  to  heart  their 
views  on  the  mathematical  curriculum. 
Practising  engineers  are  not  in  the  habit 
of  thinking  very  continuously  on  any  edu- 
cational question,  although,  of  course,  they 
must  not  confess  inability  to  respond  when 
they  are  called  upon  for  pedagogical  opin- 
ions. Every  practitioner  in  the  law  would 
doubtless  express  views  concerning  legal 
education  if  summoned  to  do  so,  but  he 
would  be  a  rash  educator  who  would  at- 
tempt to  follow  their  advice  without  much 
circumspection.  I,  myself,  prefer  to  judge 
of  the  engineer's  views  upon  educational 
matters  by  studying  his  actions  rather  than 
his   words.     The    things   engineers    "do" 


40 


SCIENCE 


may  be  taken  as  a  true  expression  of  their 
deliberate  judgment— what  they  "say"  is 
often  ill  thought  out  and  in  contradiction 
to  their  deeds.  I  therefore  prefer  to  judge 
of  the  present  needs  in  the  mathematical 
instruction  for  engineers  by  the  actual 
tendencies  that  I  observe  in  the  evolution 
of  technology  itself. 

What  are  the  great  changes  that  the 
engineering  profession  has  made  in  tech- 
nical science  in  this  country  in  the  last 
quarter  of  a  century?  The  changes  are 
quite  obvious  and  not  difficult  to  state.  In 
former  days  engineering  technology  was 
founded  chiefly  upon  current  practise 
rather  than  upon  established  principles;  it 
was  more  closely  allied  to  the  crafts  than 
to  science.  Not  only  is  that  day  past,  but 
it  is  no  longer  the  case  that  technical  sci- 
ence looks  entirely  to  pure  science  for  its 
fundamental  material.  It  has  so  grown 
that  it  is  investigating  for  itself  and,  in 
greater  and  greater  measure,  developing 
the  basal  principles  for  its  own  needs. 
There  are  very  few  American  treatises  in 
pure  science  which  will  compare  in  scien- 
tific thoroughness  with  several  treatises 
which  have  lately  issued  from  the  engineer- 
ing press.  This  is  a  very  hopeful  sign  in 
the  growth  of  knowledge— to  see  applied 
science  and  pure  science  approaching  each 
other  at  numerous  points,  so  that  it  is  in- 
creasingly difficult  to  distinguish  any  line 
of  demarcation  between  them.  In  this 
change,  science  is  not  sacrificing  any  of  its 
strength  nor  compromising  its  ideals.  It  is 
technology  that  is  changing— that  is  be- 
coming less  empirical,  more  systematic, 
more  quantitative,  more  scientific. 

With  these  well  recognized  changes  in 
applied  science  before  us,  what  should  be 
our  attitude  toward  the  mathematical  sci- 
ence that  is  necessarily  associated  with 
engineering  education?  What  is  tech- 
nology really  requiring  of  the  basal  sci- 
ences?    Judging   the   engineers   by   their 


acts  and  not  by  their  words,  what  is  the 
real  demand  that  they  are  making  of  the 
physicist,  of  the  chemist  or  of  the  mathe- 
matician ?  Is  the  demand  to  teach  physics 
or  chemistry  in  this  or  that  particular  way, 
or  is  the  demand  of  a  profounder  and  more 
radical  sort?  The  most  superficial  ob- 
servation shows  that  the  demand  is  of  the 
latter  kind.  The  engineer  in  this  twentieth 
century  is  saying  to  the  physicist,  and 
chemist,  and  mathematician :  ' '  Know  more 
science.  Discover  more  facts  in  electricity 
— in  light — in  all  properties  of  matter. 
Give  to  the  world  more  men  like  Kelvin, 
Hertz,  Helmholtz.  Fill  the  shelves  with 
ten  times  the  knowledge  we  now  have." 
These  words  more  truly  express  the  real 
pressure  that  engineers  are  putting  upon 
workers  in  pure  science,  than  do  the  words 
they  have  uttered  in  this  discussion.  As  a 
single  example,  note  that  the  great  elec- 
trical and  other  manufacturing  companies 
are  impatient  at  the  rate  at  which  pure 
science  grows,  and  large  sums  are  spent  by 
them  each  year  in  the  search  for  new 
truth  and  in  filling  up  the  gaps  in  exist- 
ing knowledge. 

The  real  demand  of  the  engineer  is  not 
for  better  instruments  or  tools  with  which 
to  do  his  work,  nor  is  the  demand  for  more 
difficult  projects  to  test  his  skill,  nor  even 
for  more  capital  with  -which  to  construct 
them.  The  real  demand  is  for  more 
knowledge,  more  science,  and  for  more  of 
the  spirit  of  science  in  technology  and  in 
technical  education.  I  take  as  my  text  a 
saying  of  Ostwald:  "Science  is  the  best 
technology."  If  we  teach  a  trade  and  not 
a  science  the  time  is  largely  wasted.  If 
we  teach  dyeing  and  not  chemistry,  the 
graduate  is  already  out  of  date  when  he 
begins  his  career,  and  he  has  not  the  funda- 
mental principles  wherewith  to  bring  him- 
self abreast  of  the  times.  I  therefore  re- 
gard it  of  greatest  importance  that  mathe- 
matics be  taught  to  engineering  students 


SCIENCE 


41 


with  real  enthusiasm  for  the  science  itself. 
It  should  be  taught  by  men  who  themselves 
are  actively  contributing  to  the  growth  of 
mathematical  science.  The  present  spirit 
of  engineering  science  is  such  that  no  in- 
structor in  any  of  the  basal  sciences  is 
satisfactory  who  does  not  see  that  it  is  his 
duty  not  only  to  teach  what  is  old,  but 
to  be  interested  in  and  to  take  an  active 
part  in  the  development  of  what  is  new. 

I  regard  of  secondary  importance  the 
particular  things  we  do  in  the  mathe- 
matical course  in  the  engineering  school. 
Different  instructors,  equally  successful, 
will  have  different  opinions.  Various 
changes  and  improvements  have  been  tried 
at  various  institutions.  At  the  University 
of  Wisconsin  we  have  made  innovations 
whenever  we  thought  it  best,  but  I  regard 
them  all  of  secondary  importance  to  the 
first  requirement  of  all,  namely,  that  we 
demand  the  right  sort  of  teachers,  and  that 
the  teaching  be  done  in  the  right  sort  of 
scientific  spirit. 

The  only  imperative  requirement  put 
upon  the  mathematics  in  engineering 
schools  that  does  not  rest  as  heavily  upon 
the  mathematics  of  the  ordinary  college 
course  is  the  demand  for  compactness.  It 
is  possible  that  there  is  some  room  in  the 
eourses  in  colleges  of  pure  science  for  the 
whims  and  fads  of  the  various  instructors, 
for  at  some  later  place  in  the  course  the 
balance  may  be  restored.  This,  however, 
is  not  true  in  a  school  of  engineering. 
There  is  very  little  room  for  the  practise 
of  fads  and  new  schemes.  It  is  easy  to 
exaggerate  the  need  of  a  special  sort  of 
subject  matter  in  mathematics  and  a  spe- 
cial class  of  problems  for  engineering  stu- 
dents. We  are  apt  to  make  some  very 
foolish  mistakes,  if  we  undertake  to  change 
too  freely  the  scientific  material  that  is 
presented  to  engineering  students.  A  good 
engineer  is  worthy  of  the  best  science  and 
the  best  instruction  that  can  be  brought  to 


him— he  himself  would  be  the  first  to 
object  if  a  different  program  were  carried 
out. 

I  have  had  a  little  experience  in  employ- 
ing engineering  graduates  in  engineering 
work.  In  the  past  ten  years  I  have  given 
employment,  in  various  capacities,  to  about 
one  hundred  and  thirty  engineering  grad- 
uates. This  work  has  been  scattered  over 
quite  a  wide  territory  and  the  men  have 
come  from  the  institutions  of  the  east,  from 
the  Pacific  Coast,  from  the  Mississippi 
Valley  and  from  the  south.  I  have  been 
able  to  judge  within  the  limits  of  my  ex- 
perience what  the  young  engineering  grad- 
uates know,  and  what  they  have  forgotten. 
I  find  it  true  that  the  boys  have  forgotten 
a  great  deal  of  the  material  they  had  in 
college,  and  that  they  have  remembered 
other  things.  They  remember  the  manual 
and  the  mechanical  things — how  to  swim, 
how  to  ride  a  horse,  how  to  fish,  how  to 
play  ball,  how  to  run  the  level,  how  to  work 
the  plane  table,  and  how  to  do  stadia  work. 
Now  what  have  they  forgotten  ?  The  men 
have  forgotten  the  intellectual  things- 
hydraulics,  electrical  science,  thermody- 
namics, etc.  The  human  mind  possesses 
an  unlimited  capacity  for  forgetting.  But 
my  experience  shows  that  the  young  men 
forget  their  hydraulics  just  as  quickly  as 
they  forget  their  mathematics  or  their 
mechanics.  The  enginee  in  the  field  ob- 
serves that  a  boy  remembers  the  right  end 
of  an  instrument  and  seems  to  be  amazed 
that  the  same  man  does  not  know  the  right 
end  of  an  integral  sign.  He  therefore 
concludes  that  the  mathematics  has  not 
been  "taught  right."  If  he  will  compare 
intellectual  things  with  intellectual  things 
he  will  find  that  a  miscellaneous  group  of 
engineers  will  pass  as  good  an  examination 
in  mathematics  ten  years  after  graduation 
as  they  would  pass  in  thermodynamics  or 
hydraulics. 

It  grates  on  me  to  hear  mathematics 


42 


SCIENCE 


spoken  of  as  a  tool.  Mathematics  is  to  the 
engineer  a  basal  science  and  not  a  tool. 
The  spirit  of  that  science  is  of  more  value 
to  the  engineer  than  the  particular  things 
that  can  be  accomplished.  The  engineer 
need  not  be  a  mathematician,  but  he  needs 
to  think  mathematically,  and,  to  my  mind, 
he  needs  the  power  of  mathematical 
thought  more  than  skill  in  manipulating 
a  few  mathematical  tools  in  mechanical 
fashion.  There  are  already  too  many  fac- 
tory-made products  turned  over  to  the  col- 
lege by  the  secondary  schools.  I  make  a 
fundamental  contrast  between  the  engineer 
with  his  mind  endowed  with  the  power  of 
creative  and  rational  design,  and  the  ar- 
tisan with  his  hands  equipped  with  tools 
for  physical  construction.  A  great  engi- 
neer must  be  trained  in  correct  seeing  and 
thinking,  and  must  have  the  power  of  rea- 
soning concerning  some  of  the  highest  ab- 
stractions of  the  human  mind.  In  this 
aspect  mathematics  is  not  a  tool — it  is  a 
basal  science. 

Chas.  S.  Slichter 
Univebsity  op  Wisconsin 

At  the  close  of  Professor  Townsend's 
address  he  urged  the  desirability  of  tech- 
nical schools  offering  more  elective  ad- 
vanced work  in  mathematics.  It  may  not 
be  out  of  place,  therefore,  for  me  to  call 
attention  to  the  fact  that  in  the  Massa- 
chusetts Institute  of  Technology  we  have 
offered  and  given,  among  others,  the  fol- 
lowing courses:  advanced  calculus,  vector 
analysis,  fourier  series,  least  squares,  the- 
ory of  surfaces,  theory  of  functions,  el- 
liptic functions,  hydrodynamics  and  dif- 
ferential equations  of  mechanics  and  phys- 
ics. Some  of  these  subjects  are  required 
in  one  or  more  of  our  courses,  but  not  in 
any  one  of  the  larger  engineering  courses, 
which  are  taken  as  the  basis  of  Professor 
Townsend's  tables.  This  elective  work, 
therefore,  while  valuable  in  many  respects, 


is  not  the  main  work  of  the  mathematical 
department. 

The  mathematical  teacher  is  in  the  engi- 
neering school  primarily  to  teach  to  stu- 
dents of  engineering  the  amount  of  mathe- 
matics which  is  necessary  to  them  for  the 
proper  understanding  and  practise  of  their 
profession.  The  object  is  to  give  the  stu- 
dent a  grasp  of  mathematical  concepts  and 
processes  through  their  use,  as  one  learns 
grammar  by  speaking  a  language.  Hence 
there  is  no  place  in  the  required  mathe- 
matics of  a  technical  school,  nor  indeed  in 
the  first  courses  in  a  college  of  liberal  arts, 
for  the  refinements  of  modern  "rigor." 
At  the  same  time  there  should  be  no  pa- 
tience with  a  loose  or  unscientific  presen- 
tation of  first  principles.  The  teacher  him- 
self must  be  thoroughly  conversant  with 
modern  thought,  else  he  will  teach  false- 
hood for  truth,  and  must  be  enthusiastic  in 
his  interest  in  his  subject,  else  he  will  fail 
to  inspire  his  pupils.  Hence  the  teacher 
of  mathematics  should  be  primarily  a 
mathematician  and  not  an  engineer.  It  is 
hard  to  find  an  engineer  who  has  any 
knowledge  of  mathematics  other  than  a 
small  fragment  which  he  habitually  uses, 
and  any  elementary  teacher  whose  instruc- 
tion goes  to  the  very  limits  of  his  knowledge 
is  sure  of  failure.  It  may,  of  course,  be 
possible  to  superimpose  a  mathematical 
training  upon  an  engineering  one,  but  in 
that  case  the  engineer  becomes  a  mathe- 
matician and  my  contention  that  mathe- 
matics should  be  taught  by  a  mathema- 
tician is  not  invalidated. 

On  the  other  hand,  the  mathematician 
should  know  something  of  the  uses  to  which 
an  engineer  wishes  to  put  mathematics. 
For  that  reason  such  meetings  as  this  are 
helpful,  but  I  must  confess  to  feeling  a 
little  disappointment  in  not  obtaining  from 
the  engineers  any  new  light  on  the  concrete 
problem  which  confronts  the  teacher  of 
mathematics  in  an  engineering  school.      I 


SCIENCE 


43 


have  met  the  same  disappointment  else- 
where in  similar  meetings.  It  has  hap- 
pened, elsewhere  if  not  here,  that  engineers 
will  tell  the  mathematicians  what  and  how 
they  should  teach,  in  apparently  total  ig- 
norance of  the  fact  that  what  the  engineer 
promulgates  as  a  new  gospel  has  been  the 
commonplace  thought  of  the  mathematician 
for  years.  This  ignorance  may  be  due  to 
the  fact  that  the  engineer  remembers  his 
own  training  of  twenty  or  thirty  years  ago 
and  does  not  know  that  improvements 
have  taken  place.  That  such  is  the 
case  may  be  seen  by  a  comparison 
of  modern  with  older  text-books.  Such 
criticism  from  the  engineers  is  amusing, 
but  another  kind  of  criticism  is  not.  I 
refer  to  the  kind  which  seizes  upon  the 
failure  of  a  student  to  have  learned  mathe- 
matics thoroughly  as  evidence  of  poor  aims 
and  inefficient  teaching  of  the  mathemat- 
ical instructor.  "We  all  know  that  students 
pass  through  our  classes  and  graduate  from 
our  schools  whose  attainments  are  not  what 
we  wish,  but  while  the  mathematical 
teacher  delivers  his  product  to  the  engi- 
neering departments  and  hears  of  his  com- 
parative failures,  the  engineering  professor 
delivers  his  product  to  the  world  and  rarely 
hears  of  the  specific  blunders  of  his  stu- 
dents. Another  unfair  criticism  is  some- 
times heard  from  the  professor  of  engineer- 
ing who  says  that  students  can  not  use  their 
mathematics,  when  the  truth  is  they  have 
simply  forgotten  some  particular  fact,  for- 
mula, or  process,  which  is  a  fad  of  that 
professor.  It  is  unfair  to  test  mathemat- 
ical training  by  tenacity  of  memory  or 
mere  quickness  in  reasoning. 

I  have  said  that  we  must  teach  our  stu- 
dents to  use  their  mathematics.  Now  in 
the  application  of  mathematics  to  a  con- 
crete problem  there  may  be  distinguished 
three  steps: 

1.  The  interpretation  of  the  data  of  the 
problem  into  mathematical  language. 


2.  The  formal  operations  upon  the  ex- 
pression or  equations  thus  obtained. 

3.  The  interpretation  of  the  results  back 
into  the  terms  of  the  original  problem. 

The  first  and  third  of  these  steps  are 
really  the  most  important,  but  there  seems 
to  be  a  popular  impression  that  the  second 
comprises  the  whole  of  mathematics.  This 
impression  is  doubtless  responsible  for  some 
criticisms  of  the  educative  value  of  mathe- 
matics. It  is  true  that  relatively  a  great 
amount  of  time  must  be  spent  in  the  class- 
room in  teaching  the  mechanical  processes 
involved  in  the  second  step,  and  many  stu- 
dents in  school  and  college  get  no  farther. 
To  object  to  the  amount  of  time  spent  in 
this  way  and  to  demand,  as  some  do,  that 
we  confine  our  time  to  teaching  general 
principles  and  applications  is  to  talk  as 
sensibly  as  a  fond  mother  who  objects  to  a 
child  beginning  his  musical  education  by 
playing  finger  exercises  instead  of  tunes. 
The  technique  of  mathematics  must  be 
learned  first,  but  the  student  who  never 
gets  beyond  the  technique  has  not  learned 
mathematics. 

The  teacher  of  mathematics  should,  then, 
use  all  possible  means  of  teaching  the  first 
and  third  of  the  above  steps  and  should 
bring  his  pupils  to  think  of  them  as  the 
real  thing.  For  that  purpose  he  should 
seek  for  applications  and  illustrations  from 
as  wide  a  range  of  subjects  as  possible. 
He  will  find  himself  handicapped,  however, 
in  using  many  problems  of  real  scientific 
or  engineering  importance  because  of  the 
ignorance  of  his  pupils,  especially  in  the 
first  year  in  the  technical  school.  To  illus- 
trate a  new  mathematical  principle  by  an 
application  to  a  science  with  which  a  stu- 
dent is  not  familiar  is  to  befog  and  not 
illumine  the  subject.  Hence  there  is  some- 
thing to  be  said  in  favor  of  some  of  the 
much-criticized  problems  of  the  older  text- 
books. To  my  mind  a  problem  is  success- 
ful if  it  causes  the  student  to  take  the  three 


44 


SCIENCE 


steps  just  enumerated  and  is  couched  in 
terms  familiar  to  the  student,  even  though 
it  may  not  be  "practical."  On  the  other 
hand,  a  type  of  problem  lately  coming  into 
use,  in  which  the  student  is  given  some 
formula  from  a  science  of  which  he  knows 
nothing,  and  is  asked  to  find,  say,  a  maxi- 
mum value,  is  as  fruitless  as  if  the  prob- 
lem were  stated  in  terms  of  x,  y  and  z, 
unless  it  may  serve  to  convince  a  sceptical 
student  that  the  matter  he  is  studying  has 
some  practical  application. 

And  this  leads  me  to  the  most  important 
thing  I  have  to  say,  and  that  is  that  after 
the  mathematical  professor  has  done  his 
utmost  to  teach  the  use  of  mathematics  the 
engineering  professor  must  take  up  and 
complete  his  work.  I  doubt  if  any  one 
really  learned  the  use  of  mathematics  in  a 
first  course.  Facility  in  using  mathematics 
comes  from  actual  use  and  not  from  the 
solution  of  illustrative  examples.  In  the 
course  in  mathematics  the  student  expects 
his  problem  to  be  solved  mathematically 
and  has  his  mind  alert  to  find  the  solution, 
and  that  too  with  mathematical  principles 
fresh  in  his  mind.  In  a  course  in  engi- 
neering, his  point  of  view  has  widely 
changed.  The  practical  problem  has  now 
his  main  interest,  mathematical  concepts 
are  in  the  background,  and  he  often  fails 
to  see  the  possibility  of  using  mathematical 
principles  until  he  is  trained  to  do  so  by 
the  professor  of  engineering.  If  the  pro- 
fessor, through  lack  of  knowledge  or  lack 
of  interest,  avoids  the  use  of  mathematics, 
the  student  will  soon  lose  the  little  he  has 
learned. 

In  other  words,  the  mathematical  train- 
ing of  a  student  is  not  complete  when  he 
leaves  the  department  of  mathematics.  It 
is  possible  that  better  results  could  be  ob- 
tained if  the  mathematical  department  had 
more  time,  say  for  a  course  in  applications 
of  mathematics  to  miscellaneous  problems. 
But,  as  a  rule,  in  our  technical  schools  the 


department  of  mathematics  is  allowed 
barely  time  to  teach  the  necessary  tech- 
nique with  what  illustrations  and  applica- 
tions can  be  squeezed  in.  Hence  the  math- 
ematical department  delivers  to  the  engi- 
neering department  an  unfinished  product 
and  it  is  the  engineer's  duty  to  teach  the 
student  to  use  the  mathematics  he  has 
learned.  Unfortunately,  the  professor  of 
engineering  is  too  often  a  poor  mathema- 
tician and  avoids  this  duty. 

One  of  the  hardest  things  a  student  has 
to  do  is  to  combine  two  different  domains 
of  knowledge,  each  somewhat  unfamiliar, 
so  that  he  may  work  freely  in  both  at  once, 
using  each  as  a  help  in  the  other.  It  is 
this  difficulty  which  makes  analytical  geom- 
etry traditionally  hard,  and  which  the  stu- 
dent meets  again  when  he  studies  any  form 
of  applied  mathematics.  It  is  partly  to 
help  overcome  this  difficulty  that  we  have 
just  made  a  rearrangement  of  our  mathe- 
matical instruction  in  the  Massachusetts 
Institute  of  Technology.  We  no  longer 
have  courses  in  algebra,  analytic  geometry 
and  differential  and  integral  calculus,  but 
have  combined  these  into  one  "course  in 
mathematics"  extending  through  two 
years.  Into  this  course  the  elements  of 
analytic  geometry  and  of  calculus  are  in- 
troduced early  and  continued  late.  We 
hope  thus  to  give  these  principles  more 
time  to  become  completely  domiciled  in  the 
student's  mind.  We  have  also  been  en- 
abled to  carry  out  two  principles :  the  first 
is  to  introduce  no  subject  until  some  use 
is  to  be  made  of  it,  and  the  second  to 
handle  each  problem  by  the  method  best 
adapted  to  it,  rather  than  by  the  methods 
of  the  particular  branch  of  mathematics 
which  one  might  at  the  moment  be  study- 
ing under  the  old  classification.  We  hope 
in  this  way  to  increase  the  efficiency  of  our 
mathematical  teaching. 

F.  S.  Woods 

Massachusetts  Institute  of  Technology 


SCIENCE 


45 


The  program  shows  three  standpoints 
from  which  discussion  is  to  emanate.  I 
occupy  no  one  of  them.  It  is  true  I  have 
had  some  engineering  practise,  but  I  can 
not  be  termed  a  practising  engineer.  I 
have  had  charge  of  mathematics  for  engi- 
neering students  in  two  engineering  col- 
leges, but  for  nearly  a  decade  now  I  have 
not  met  students  in  mathematics;  and, 
indeed,  I  have  taught,  all  told,  but  an  insig- 
nificant amount.  I  am  in  somewhat  close 
touch  with  engineering  students,  but  they 
belong  to  a  particular  field,  namely, 
mining,  which  is  possibly  less  dependent 
on  mathematics  than  are  other  branches 
of  engineering.  My  view-point  is,  there- 
fore, somewhat  of  a  compromise  or  average 
of  the  three  specified  in  the  announcement. 

The  present  discussion  seems  to  me  sig- 
nificant. It  may  bring  forth  results.  In 
fact  it  seems  to  have  had  some  immediate 
consequences.  Last  evening  after  the  din- 
ner I  heard  a  very  clever  mathematician 
admit  that  he  felt  really  humble,  and  I 
heard  a  well-known  engineer  say  that  to 
his  great  surprise  some  mathematicians  had 
a  human  side.  I  asked  a  pure  mathema- 
tician sitting  near  me  to  show  me  his  hu- 
man side,  but  he  only  shrugged  his  shoul- 
ders. Perhaps  he  was  not  yet  sufficiently 
humbled. 

This  occasion  appears  to  me  to  be  signi- 
ficant, but  as  showing  conditions  which 
exist  rather  than  as  forecasting  future 
changes.  It  is  a  symptom  of  the  approach 
—the  arrival,  perhaps— of  healthful  condi- 
tions rather  than  a  cause.  It  may,  of 
course,  in  its  turn  become  a  cause,  and 
operate  toward  good  results.  That  is  not 
so  certain.  At  the  moment  it  indicates 
conditions  surrounding  the  teaching  of 
mathematics  to  engineering  students,  in- 
cluding the  relations  between  the  teachers 
of  mathematics  and  those  of  engineering 
which  have  been  the  growth  of  many  years. 
Those  young  and  virile  gentlemen  whom  we 


all  delight  to  honor,  the  Woodwards,  have 
been  striving  for  decades  to  bring  about  a 
closer  relation  between  the  teaching  of 
mathematics  and  the  subsequent  study  of 
practise  of  engineering.  Ten  years  ago  at 
the  Toronto  meeting  of  the  Society  for 
Promotion  of  Engineering  Education  I  pre- 
sented a  paper  looking  to  this  end.2  There 
are  gentlemen  here  present  who  discussed 
that  paper  and  who  may  perhaps  recall  the 
remarkable  unanimity  between  the  teachers 
of  mathematics  and  those  of  engineering  as 
to  the  results  most  to  be  desired  in  teaching 
mathematics  to  engineering  students,  and, 
indeed,  as  to  the  best  available  methods  for 
producing  such  results.  This  movement  is 
old.  Most  of  the  ideas  which  have  been 
brought  out  here  were  first  conceived  a 
long  time  since.  Nevertheless,  it  is  good 
to  get  together  and  talk  them  over,  and 
such  discussions  may  result  in  help  to  the 
individual  teacher. 

We  have  heard  here  much  of  the  ideal 
which  the  engineering  school  should  set 
before  itself,  but  it  might  well  be  asked 
what  problem  is  presented  first  to  the 
school  as  a  matter  of  fact?  President 
Woodward  put  it  in  part  when  he  spoke  of 
the  difficulty  of  getting  the  right  men  in 
the  schools  when  operators  are  so  eager 
for  good  men  and  are  competing  on  the 
basis  of  ' '  so  much  per  month. ' '  And  what 
do  the  employers  demand?  They  call  for 
men  who  can  do  something,  men  who  can 
think  in  a  logical  and  common-sense  way, 
but,  withal,  when  they  leave  the  school  can 
be  put  to  some  immediate  use.  The  first 
problem  confronting  the  engineering  col- 
lege is  how  to  meet  this  demand,  for  the 
demand  must  be  met  in  some  degree  at 
least  or  the  college  will  cease  to  train  men. 

It  is  inevitable  that  the  character  of  this 
demand  shall  influence  largely  what  the 
school  must  do.     The  call  is  not  for  men 

1  See  Proceedings  of  Society  for  Promotion  of 
Engineering  Education,  Vol.  V.,  1897,  p.  139. 


46 


SCIENCE 


highly  trained  in  mathematics,  however 
much  we  may  feel  it  ought  to  be.  It  is 
for  men  who  know  well  a  little  mathe- 
matics, and  who  can  do  something  with  it, 
who  can  use  it  "as  a  tool."  And,  however 
obnoxious  that  expression  may  be  to  a 
mathematical  teacher,  he  who  forgets  or 
disregards  the  fact  which  lies  behind  it 
will  surely  weaken  his  instruction  of  engi- 
neering students. 

I  do  not  defend  the  specification  of  the 
employer,  I  point  to  the  fact  with  which  we 
must  deal.  Personally  I  am  inclined  to 
find  fault  with  it,  but  the  matter  rests 
largely  in  the  hands  of  the  practising  engi- 
neer. He,  though  he  often  objects  to  the 
college  product,  is  to  a  great  extent  re- 
sponsible for  its  general  make-up.  In  the 
long  run  and  within  reasonable  limits  he 
can  have  what  he  wants.  Sometimes  he  is 
inclined  to  require  too  much  technical 
knowledge  on  the  part  of  the  graduate. 
His  brother  teaching  in  the  college  in  order 
to  meet  his  requirement  says  to  the  teacher 
of  mathematics  I  must  have  those  students 
ready  earlier  with  their  mathematics.  This 
fact,  together  with  the  general  tendency  in 
the  colleges  to  raise  the  standards,  causes 
the  mathematical  training  to  be  crowded 
into  the  first  year  and  a  half  or  two  years, 
when  the  student  is  least  mature.  More 
of  it  is  being  pushed  back  to  the  second- 
ary school,  and,  in  turn,  into  the  grades. 
Mathematical  concepts  are  difficult,  and 
with  President  Woodward  I  am  inclined  to 
think  we  are  demanding  too  much,  and 
calling  for  it  too  soon.  Covering  less 
ground  and  at  a  slower  pace  will  help  to 
make  better  engineers. 

The  student  comes  to  the  engineering 
school  with  the  notion  that  he  is  to  be 


filled  up  with  a  lot  of  technical  knowl- 
edge, the  items  of  which  will  be  used  by 
him  when  he  is  a  practising  engineer.     He 
seems  unable  to  comprehend  that  he  is  in 
college  to  acquire  mastery  over  his  own 
powers.     He  is  eager  for  useful  facts  and 
of   course   he   forgets  most   of   those   he 
learns  not  a  great  while  after  leaving  col- 
lege.    The   forgetting   is  to  be   assumed. 
Under  such  conditions  the  task  before  the 
teacher  of  mathematics,  and  quite  as  well 
before  the  teacher  of  engineering,  is  to  do 
his  utmost  to  train  his  student  to  think 
logically  and  accurately  about  things.     To 
this  end  there  seems  to  me  nothing  so  effi- 
cient as  the  solution  of  a  large  number  of 
carefully  chosen  problems.     Indeed  what  is 
one's  life,  if  it  be  active,  except  meeting  a 
never  ending  succession  of  problems  which 
must  be  solved  if  success  is  to  be  gained? 
If  you  can  teach  your  student  to  take 
vigorous  hold  of  a  problem,  to  first  as- 
semble all  the  facts  which  bear  on  the 
question,   then  from  the  facts  to  reason 
logically  to  a  sound  and  safe  conclusion, 
you  have  started  him  well  whether  his  aim 
be  engineering  or  otherwise. 

Of  transcendent  importance  is  the 
teacher,  his  personality,  his  attitude  toward 
his  work,  his  knowledge  of  his  students, 
not  as  a  class,  but  of  each  as  a  human 
being.  If  we  can  procure  the  teacher  who 
can  idealize  his  work,  who  can  show  sus- 
tained enthusiasm  for  it  and  perform 
cheerfully  the  drudgery  we  heard  men- 
tioned a  few  minutes  ago,  we  can  safely 
leave  detailed  methods  to  him.  Whatever 
methods  such  a  man  adopts  in  the  class- 
room are  likely  to  be  effective. 

Fred  W.  McNair 

Michigan  Coixege  of  Mines 


SCIENCE 


47 


THE   TEACHING   OF   MATHEMATICS   TO 
STUDENTS    OF   ENGINEERING1 

WHAT  IS  NEEDED  IN  THE  TEACHING  OP 
MATHEMATICS  TO  STUDENTS  OP  ENGINEER- 
ING? (a)  RANGE  OP  SUBJECTS;  (6)  EX- 
TENT IN  THE  VARIOUS  SUBJECTS;  (c) 
METHODS  OP  PRESENTATION;  (d)  CHIEF 
AIMS. 

By  Calvin  M.  Woodward,  Professor  of 
Mathematics  and  Applied  Mechanics, 
and  Dean  of  the  School  of  Engineering 
and  Architecture,  Washington  Univer- 
sity. 

I  want  to  emphasize  the  point  which  Mr. 
Scott  has  just  touched  on,  and  that  is  that 
we  often  attempt  too  early  to  teach  the 
subjects  that  require  mature  and  reflect- 
ing minds.  I  want  to  tell  you  a  story,  a 
true  biography  of  some  one  you  all  know 
of.  He  went  through,  in  the  city  of  New 
York,  the  whole  range  of  mathematics,  in- 
cluding analytic  geometry  and  calculus. 
He  learned  his  formulae  and  definitions 
and  "passed"  in  some  manner,  but,  he  told 
me,  he  did  not  know  anything  about  them. 
He  believed  he  was  a  dunce,  and  whenever 
he  was  required  to  make  an  intelligible 
demonstration,  he  could  not  do  it;  his 
teachers  and  his  parents  concluded  that  he 
was  a  dunce  in  mathematics,  and  could 
never  do  anything  in  it.  He  would  have 
gone  through  life  with  that  notion,  if  some 
one  had  not  offered  him  an  appointment 
to  West  Point.  He  doubted  his  ability  to 
pass  the  entrance  examination  in  arith- 
metic; but  his  friends  advised  him  to  get 
an  arithmetic  and  study.  He  bought  a 
bcok  and  sat   down  and  read  the  book 

1  General  discussion  following  the  presentation 
of  four  formal  papers  (see  Science,  July  17,  26, 
31,  1908),  and  of  the  eight  prepared  discussions 
(see  Science,  August  7  and  28,  1908).  Presented 
before  Sections  D  and  A  of  the  American  Associa- 
tion for  the  Advancement  of  Science  and  the  Chi- 
cago Section  of  the  American  Mathematical  So- 
ciety, at  the  Chicago  meeting,  December  31,  1907. 


through,  and  to  his  astonishment  he  found 
it  easy.  He  passed  his  examination  with 
flying  colors.  He  entered  West  Point  and 
graduated  at  the  head  of  his  class  in 
mathematics,  and  is  now  at  the  head  of  a 
high  grade  technical  school.  If  it  had  not 
been  for  the  opportunity  of  going  again 
over  his  whole  course  of  mathematics,  he 
would  have  gone  to  his  grave  thinking  he 
had  no  capacity  for  mathematical  analysis. 
That  comes  from  poor  or  premature  teach- 
ing. 

I  am  opposed  to  putting  college  mathe- 
matics in  high  schools.  Those  young 
people  may  get  a  glimmer  of  it,  but  they 
get  false  impressions  from  it  which  are 
hard  to  remove.  I  have  been  teaching 
mathematics  for  forty  years  or  more,  and 
have  been  teaching  applied  mechanics  for 
the  same  time.  I  taught  Rankine  for 
twenty-five  years.  It  has  always  been  my 
duty  and  my  privilege  to  make  my  stu- 
dents see  what  mathematics  was  good  for. 
And  I  want  to  defend  the  teachers  of  high 
school  and  freshman  mathematics  from 
what  I  think  is  unjust  criticism.  It  is 
charged  they  do  not  make  their  students 
understand  what  mathematics  is  good  for. 
It  is  simply  impossible  for  them  to  do  so, 
as  I  can  do  in  mechanics.  A  man  is  very 
fortunate  who  can  teach  mathematics  and 
then  show  what  it  is  good  for.  I  am  old 
enough  to  quote  a  little  of  my  early  experi- 
ence. I  am  led  to  it  by  something  Pro- 
fessor Swain  said  in  regard  to  mental 
processes.  There  is  nothing  so  valuable  to 
mathematical  success  as  a  clear  grasp  of 
fundamental  principles.  When  I  was  pre- 
paring for  college  I  gave  all  my  time  to 
Latin  and  Greek.  I  had  done  all  my 
freshman  mathematics  and  was  reputed  to 
be  strong  on  that  branch,  when  a  new 
teacher  came  into  the  school  who  said, 
"Here's  a  new  book  in  intellectual  arith- 
metic, and  I  would  like  to  have  every  stu- 
dent in  the  school  go  through  it."    It  was 


84 


SCIENCD 


fun  for  me,  of  course,  but  I  went  through 
the  book  from  A  to  Z;  no  other  mathe- 
matics that  I  ever  studied  did  me  so  much 
good.  The  teacher's  maxim  was,  "Take 
hold  of  the  thread  at  the  right  end." 
That  was  the  secret  of  his  splendid  teach- 
ing. I  have  applied  that  maxim  to  every 
branch  of  mathematics  I  have  ever  studied 
or  taught.  I  have  learned  to  take  hold  of 
mathematics  at  the  right  end,  and  in  a 
measure  I  have  taught  my  students  to  do 
so. 

By  B.  F.  Groat,  Professor  of  Mechanics 
and  Mathematics,  School  of  Mines,  Uni- 
versity of  Minnesota. 
Most  of  the  speakers  have  stated  that 
what  they  were  about  to  say  had  already 
been  said  by  preceding  speakers.  I  am 
going  to  try  to  state  a  general  principle  I 
have  not  heard  clearly  put  since  I  came 
here.  During  the  lunch  hour  Professor 
Slaught  said  that  he  had  not  heard  a 
single  general  pedagogical  principle 
brought  out.  I  am  going  to  take  the 
honor  to  myself,  to  give  expression  to  what 
seems  to  me  to  be  a  general  educational 
principle. 

Mathematics  is  mathematics  and  engi- 
neering is  engineering.  There  is  just  as 
much  art,  science  or  principle  in  the  teach- 
ing of  mathematics  as  there  is  in  the  teach- 
ing of  engineering  and  these  two  subjects 
should  be  distinguished,  separated  and 
kept  separate.  If  you  are  going  to  teach 
engineering  you  must  teach  the  pure  prin- 
ciples. If  you  are  going  to  teach  mathe- 
matics you  have  got  to  teach  pure  mathe- 
matics. Let  it  be  pure  or  applied  mathe- 
matics, it  is  the  principle  involved  which 
must  be  taught.  If  this  rule  is  not  ad- 
hered to  we  shall  find  ourselves  teaching 
something  different  from  that  which  it  was 
intended  to  teach. 

The  principle  is  that  the  technical 
courses  in  our  engineering  schools  must  be 


separated  from  our  general  educational 
courses.  The  technical  courses  are  for  the 
purpose  of  fitting  the  man  for  a  special 
life  work  which  is  to  come  later  on.  The 
general  education  which  he  should  have, 
by  way  of  preparation,  should  precede  his 
technical  course  as  far  as  possible. 

The  straight  technical  course  should  be 
given  as  a  course  of  two  years  extent, 
while  the  general  and  preparatory  sub- 
jects should  precede  in  a  three-  or  four- 
year  course. 

The  University  of  Minnesota  has  adopted 
a  five-year  engineering  course.  This  is 
along  the  lines  I  am  recommending  and  I 
prophesy  that  it  will  soon  be  extended  to 
other  schools  and  separated  into  two  parts. 

Let  your  professor  of  engineering  teach 
engineering  and  your  professor  of  mathe- 
matics teach  mathematics.  That  is  the 
general  pedagogical  principle  I  want  to 
announce. 

By  C.  S.  Howe,  President,  Case  School  of 

Applied  Science. 

I  have  been  very  much  interested  in  the 
discussion  of  this  subject  because  for 
thirteen  years  I  was  a  professor  of  mathe- 
matics in  an  engineering  school  and  dur- 
ing the  past  five  years  I  have  been  en- 
deavoring to  reconcile  the  differences  be- 
tween professors  of  mathematics  and  pro- 
fessors of  engineering.  One  thing  in  this 
discussion  which  strikes  me  as  very  pecul- 
iar is  the  sad  lack  of  knowledge  displayed 
by  the  engineering  professors  as  to  what  is 
being  done  in  mathematics  in  their  own 
schools.  I  believe  from  my  experience  and 
from  what  I  have  seen  in  other  institutions 
that  the  professors  of  mathematics  are 
teaching  mathematics  most  admirably  as 
mathematics,  but  they  are  not  teaching 
mathematics  as  a  department  of  engineer- 
ing. I  do  not  believe  that  mathematics 
should  be  taught  as  a  department  of  engi- 
neering.    Mathematics  is  a  science  in  itself 


SCIENCE 


49 


and  should  be  taught  by  specialists  in  that 
science  if  our  students  are  to  be  trained 
in  the  proper  way.  The  professor  of 
mathematics  has  two  duties  to  perform. 
One  is  to  teach  his  students  the  principles 
of  mathematics— that  is,  to  teach  them  to 
reason  and  to  understand  why  certain 
processes  are  right  and  why  others  are 
wrong.  The  student  must  also  be  taught 
how  to  use  his  mathematics  so  that  he  can 
solve  any  problem  as  soon  as  that  problem 
is  expressed  in  mathematical  terms. 
Another  duty  of  the  mathematician  is  to 
teach  the  student  to  be  exact.  Unless  the 
engineer  is  exact,  unless  he  can  obtain 
definite  and  reliable  results  in  his  engi- 
neering work,  he  can  not  succeed  in  his 
profession.  This  accuracy  must  be  very 
largely  taught  in  the  mathematical  depart- 
ment and  much  of  the  time  and  care  be- 
stowed upon  classes  is  for  the  purpose  of 
accomplishing  this  result. 

I  believe  also  that  the  professors  of  engi- 
neering are  teaching  engineering  thor- 
oughly and  well.  The  difficulty  which  we 
are  discussing  to-day  is  not  in  the  teaching 
of  mathematics  alone  nor  in  the  teaching 
of  engineering  alone,  but  in  the  connection 
between  the  two.  The  technical  student 
is,  I  believe,  taught  pure  mathematics  well, 
but  when  he  enters  the  class  in  engineering 
he  finds  that  he  has  to  deal  with  mathe- 
matics under  a  new  form— that  is,  the  par- 
ticular engineering  subject  he  is  studying 
must  be  translated  into  mathematical 
terms  and  this  is  where  he  frequently 
meets  with  great  difficulty.  The  student 
in  algebra  who  has  learned  to  solve  equa- 
tions of  the  first  degree  may  have  great 
difficulty  with  problems  involving  equa- 
tions of  the  first  degree  because  he  has  not 
learned  to  state  the  problems  in  mathe- 
matical language.  So  the  student  who 
begins  electrical  work  finds  certain  prob- 
lems containing  known  and  unknown 
quantities,  but  not  yet  expressed  in  mathe- 


matical terms.  Now  I  can  not  believe  that 
it  is  the  duty  of  the  professor  of  mathe- 
matics to  teach  the  student  to  express  prob- 
lems in  the  various  branches  of  engineer- 
ing in  the  form  of  equations  or  other 
mathematical  terms.  In  order  to  do  this 
it  would  be  necessary  for  him  to  under- 
stand all  the  various  branches  of  engineer- 
ing and  it  is  manifestly  impossible  for 
him  to  do  this.  The  professor  of  civil 
engineering  understands  the  problems  of 
that  subject  and  he  should  show  the  stu- 
dent in  his  department  how  to  express 
these  problems  in  such  terms  that  the  stu- 
dent can  deal  with  them  mathematically. 
The  same  may  be  said  of  each  of  the  de- 
partments of  engineering.  When  the  pro- 
fessors of  engineering  have  taught  their 
students  to  state  the  problems  of  their  own 
departments  in  mathematical  language, 
then  the  student  who  has  had  the  course 
in  mathematics  ought  to  be  able  to  solve 
the  problems,  and  if  he  can  not  he  has  not 
been  taught  his  mathematics  thoroughly  or 
so  much  time  has  elapsed  since  he  studied 
the  subject  that  he  has  forgotten  some 
parts  of  it. 

Again,  I  believe  that  the  professor  of 
engineering  should  ascertain  in  a  general 
way  how  mathematics  is  being  taught  in 
his  institution  and  in  just  what  form  the 
student  is  using  certain  terms  so  that  he 
may  express  his  own  problems  in  a  way 
familiar  to  the  student.  If,  for  instance, 
in  calculus  the  mathematical  department 
has  been  using  derivations,  the  professors 
of  engineering  in  writing  their  problems 
should  use  differential  coefficients  and  not 
attempt  to  express  problems  in  terms  of 
differentials.  I  know  from  experience  that 
many  professors  of  engineering  do  not  do 
this  and  their  students  are  confused  by  a 
difference  of  terms  and  not  by  a  lack  of 
knowledge  of  the  subject.  It  is  evident 
that  the  professors  of  engineering  must 
conform  to  the  methods  of  the  department 


50 


SCIENCE 


of  mathematics  because  the  department  of 
mathematics  can  use  but  one  method  while 
the  five  or  more  departments  of  engineer- 
ing might  have  several  different  methods. 
It  is  obvious,  then,  that  for  the  sake  of 
simplicity  one  method  must  be  used  and 
that  method  must  be  the  method  of  the 
department  of  mathematics. 

I  also  believe  that  the  professor  of 
mathematics  should  occasionally  confer 
with  the  professors  of  engineering  in  order 
to  find  out  from  them  just  what  mathe- 
matical subjects  engineering  students  are 
weak  in  and  what  subjects  it  is  especially 
desirable  to  have  them  well  trained  in  and 
to  see  that  his  students  are  taught  these 
things.  Friendly  conferences  between  the 
departments  are  of  great  value  and  should 
be  encouraged  by  both  the  mathematicians 
and  the  engineers. 

By    Clakence   A.    Waldo,    Professor   of 

Mathematics,  Purdue  University. 

In  the  table  of  hours  for  mathematics 
in  the  various  institutions  cited  by  Pro- 
fessor Townsend,  the  largest  total  stands 
against  Purdue.  Also  a  whole  semester  is 
assigned  to  trigonometry.  Both  of  these 
conditions  are  in  a  measure  due  to  the  fact 
that  we  have  recently  passed  through  a 
transitional  period  in  which  for  engineers 
solid  geometry  has  been  relegated  to  the 
secondary  schools.  The  first  semester  was 
formerly  divided  between  solid  geometry 
and  trigonometry.  Now  it  is  wholly  given 
to  the  latter,  while  the  second  semester  is 
set  aside  to  college  algebra.  Experience 
shows  that  for  the  ordinary  student  college 
algebra  is  more  difficult  than  trigonometry 
and  this  determines  their  order  in  our 
program. 

Placing  trigonometry  first  and  giving  it 
so  much  time  has  developed  with  us  several 
interesting  facts. 

1.  Being  easy  to  understand  and  having 


interesting   applications,  it   naturally  fol- 
lows secondary  work. 

2.  While  trigonometry  is  easy  to  under- 
stand, yet  to  acquire  facility  in  its  use  and 
absolute  mastery  over  it  as  a  fundamental 
science  requires  close  and  long-continued 
study,  yet  the  student,  ambitious  to  become 
an  engineer,  quickly  sees  that  he  must  have 
facility  in  this  subject  and  mastery  over  it. 

As  a  subject  of  study,  therefore,  at  the 
beginning  of  a  young  man's  college  career 
it  is  well  adapted  to  give  power  and  to 
instill  habits  of  thoroughness,  application, 
concentration  and  mastery. 

3.  Engineers  have  been  recommending 
that  a  generous  amount  of  time  shall  be 
given  to  trigonometry,  at  the  expense  of 
the  calculus  if  necessary. 

4.  The  subject  is  used  to  review  and 
emphasize  much  of  the  preparatory  mathe- 
matics, while  it  is  also  used  to  clear  the 
way  for  that  which  is  to  come. 

Another  peculiarity  in  which  Purdue 
stands  almost  alone  we  are  quite  prepared 
to  defend.  We  do  not  crowd  the  pure 
mathematical  work  into  the  first  two  years, 
much  less  into  the  first  year,  but  give  it 
an  hour  less  in  the  second  year,  than  the 
first,  yet  at  the  outset  of  the  third  year, 
with  his  first  course  of  calculus  fairly 
mastered,  we  have  the  student  well  pre- 
pared to  begin  attack  upon  theoretical  me- 
chanics and  kindred  subjects.  However, 
with  two  hours  a  week  during  junior  year 
devoted  to  the  further  exploration  of  the 
calculus  carried  on  side  by  side  with  its  ap- 
plication to  studies  of  a  nature  more  or 
less  professional,  like  thermodynamics,  the 
student  is  likely  to  come  finally  into  living 
contact  with  calculus  ideas.  Through 
three  years,  then,  mathematical  ideas  are 
held  persistently  and  prominently  before 
the  mind  of  the  student,  so  that  at  the  end 
of  that  time  the  mental  change  which  I 
call  the  mathematical  transformation  is 
quite  complete.     If  you  are  intent  upon 


SCIENCE 


51 


making  a  physical  transformation  by 
which  a  weak  man  becomes  robust  and 
powerful,  you  give  one,  two  or  three  years 
for  the  muscles  to  grow  and  the  chest  to 
expand  through  long-continued  and  sys- 
tematic exercise.  Similarly  the  average 
student  does  not  become  habitually  mathe- 
matical and  exact  in  his  thinking  unless 
you  give  him  careful  direction  and  devote 
plenty  of  time  to  his  development.  The 
man  who  uses  his  memory  and  copies 
slavishly  must  disappear.  In  his  place 
must  stand  the  man  of  trained  intellect, 
thoughtful,  persistent,  rich  in  expedients, 
powerful  in  attack.  To  produce  him  there 
are  on  the  mathematical  side  two  in- 
dispensable  requisites,  thoroughness  in  the 
fundamentals,  and  a  sufficient  time  to 
make  the  mathematical  attack  of  a  prob- 
lem habitual  and  natural,  and  to  give  such 
a  control  of  and  power  in  the  use  of  the 
tools  of  mathematics  that  the  solution  of  a 
problem  of  average  difficulty  shall  be  easy 
and  pleasurable. 

In  the  required  mathematical  part  of  the 
engineering  courses  at  Purdue  these  are  the 
considerations  that  determine  the  distribu- 
tion of  the  work  in  the  four-year  program, 
and  all  of  the  time  we  are  teaching  not 
alone  the  particular  subject  that  happens 
to  be  named  in  the  curriculum— but  mathe- 
matics. 

Some  years  ago  it  was  my  fortune  to 
study  descriptive  geometry  under  Marx 
and  Von  Derlin  in  Munich.  They  taught 
their  subject  from  the  standpoint  of  the 
mathematician  rather  than  that  of  the 
draftsman.  They  made  their  students 
visualize  geometric  form  in  space  and  by 
the  use  of  that  power  discover  methods  of 
solving  on  paper  synthetic  problems  of 
much  difficulty.  The  German  schools 
teach  descriptive  geometry  as  a  mathe- 
matical subject,  the  American  schools  as  a 
body  of  problems  to  be  solved  by  rule  on 
the  drawing  board.     The  former  method 


makes  descriptive  geometry  the  finest  dis- 
cipline of  the  four  years'  course;  from 
the  other  method  little  educational  benefit 
arises.  Some  years  ago  at  the  Rose  Poly- 
technic, where  for  a  time  we  taught  &e- 
scriptive  geometry  in  the  German  way,  it 
was  not  unusual  to  meet  students  who 
declared  enthusiastically  that  they  got 
more  real  good  from  this  subject  than  from 
anything  else  in  their  entire  course. 

I  would  ask  the  new  committee  to  inquire 
how  and  by  whom  descriptive  geometry 
should  be  taught? 

By  C.  B.  Williams,  Professor   of  Mathe- 
matics, Kalamazoo  College. 
The    teachers    of    mathematics    in    the 
small  colleges  of  the  middle  west  are  pre- 
paring many  men  for  work  in  the  better 
technical   schools.     From    our   standpoint 
there  is  substantial  agreement  between  the 
two  representatives  of  the  Massachusetts 
Institute  of  Technology  (Professors  Wood 
and  Swain).     They  expressed  themselves 
so  differently  that  one  might  easily  fail  to 
see  how  closely  they   agree.    Both  want 
longer    and    stronger    courses    in    mathe- 
matics in  the  secondary  schools.     I  would 
like  to  know  the  college  teacher  of  mathe- 
matics  who    does   not   agree   with    them. 
They  want  more  mathematics  taught  and 
to  have  it  taught  better,  to  have  longer  and 
more  consecutive  mathematical  courses  in 
the  secondary   and  primary  schools.     In 
other  words,  the  faculties  of  the  technical 
schools  and  colleges  are  working  toward 
the  same  end,  that  is,  to  have  more  effective 
courses  in  primary  and  secondary  mathe- 
matics so  that  college  students  can  do  more 
and   better   mathematical   work.       If   we 
could  have  properly  prepared  students,  we 
could  turn  out  the  kind  of  men  the  better 
technical  schools  should  have. 

The  engineers  and  teachers  of  engineer- 
ing have  insisted  that  the  most  necessary 
qualification  for  a  real  engineer  is  that  he 


52 


SCIENCE 


should  be  able  to  realize  his  mathematics, 
to  ' '  think  mathematically, ' '  as  they  express 
it.  The  mathematicians  want  the  same 
thing.  We  are  trying  to  make  use  of  and 
to  train  the  faculty  of  geometric  intuition, 
to  emphasize  the  functional  notion  and  to 
develop  functional  thinking.  There  is 
substantial  agreement  that  the  best  way 
to  do  this  is  through  geometry,  with  per- 
haps some  help  from  elementary  me- 
chanics. It  is  true  that  sometimes  we  are 
tempted  to  use  too  big  and  complicated 
machines  for  little  problems,  but  this  is 
only  because  we  are  attempting  to  develop 
methods  powerful  enough  to  solve  big 
problems. 

By  J.  B.  Webb,  Professor  of  Mathematics 

and  Mechanics,  Stevens  Institute. 

Every  practical  problem  requiring 
mathematics  for  its  solution  consists  of 
three  parts: 

(a)  An  Analysis,  which  resolves  the 
problem  into  its  elements,  examines  these 
in  the  light  of  natural  laws,  rejects  unim- 
portant ones  and  defines  the  relations  exist- 
ing between  those  upon  which  the  solution 
depends.  This  involves  the  adoption  or 
discovery  of  methods  of  measuring  the 
elements,  so  that  they  may  be  expressed 
quantitatively  by  symbols,  and  of  the  re- 
duction of  the  relations  between  them  to 
the  standard  mathematical  forms  of  ex- 
pression. The  result  is  a  mathematical 
statement  of  the  problem  by  one  or  more 
equations. 

(&)  A  solution  of  the  equations  by  which 
the  relations  sought  for  between  the  quan- 
tities are  clearly  expressed  or  the  quanti- 
ties put  in  proper  form  to  have  their  values 
calculated. 

(c)  The  interpretation  of  the  result, 
which  involves  a  translation  of  the  same 
from  the  mathematical  language  in  which 
it    has    been    obtained    into    the    original 


language  of  the  problem  and  a  discussion 
of  the  practical  bearings  of  the  same. 

In  conversation  with  a  fellow  mathe- 
matician at  this  meeting  he  surprised  me 
by  saying  that  he  expected  a  problem  to 
be  put  into  mathematical  language  before 
it  was  submitted  to  him  and  I  presume  he 
did  not  feel  bound  to  interpret  his  results. 
Now  if  "pure  mathematicians"  regard 
practical  problems  in  this  way,  engineers 
and  other  practical  men  have  just  cause 
for  finding  fault  with  "pure  mathe- 
matics," and  to  teach  mathematics  in  this 
way  is  to  render  it  valueless  to  most  stu- 
dents. Personally  I  should  refuse  to 
undertake  a  problem  unless  I  made  the 
analysis  and  interpretation  as  well  as  the 
solution. 

In  many  if  not  in  most  problems  the 
analysis  and  interpretation  are  the  main 
parts.  They  require  a  broad  knowledge  of 
practical  conditions  and  of  other  sciences 
and  are  far  more  interesting  than  the  mere 
solution,  especially  as  they  often  bring  into 
play  a  large  amount  of  ingenuity  and  in- 
vention, as  well  as  imagination  and  judg- 
ment. A  mathematician  who  can  not 
make  the  analysis  and  interpretation  of  a 
problem  is  not  to  be  trusted  with  the  solu- 
tion and  an  engineer  who  is  fully  compe- 
tent to  make  them  had  better  undertake 
the  solution  himself  or  put  the  whole  prob- 
lem into  the  hands  of  a  mathematician 
fully  competent  to  undertake  it. 

There  is  no  excuse  for  a  "pure  mathe- 
matician" remaining  ignorant  of  the  prac- 
tical side  of  the  problems  he  teaches,  and 
his  mathematics  will  not  be  interesting  or 
trustworthy.  Let  him  cultivate  the  ac- 
quaintance of  the  truly  educated  engineer, 
who  will  be  only  too  glad  to  discuss  prob- 
lems with  him  and  give  him  all  the  prac- 
tical information  he  needs.  But  there  are 
too  many  engineers  who  are  not  truly  edu- 
cated and  who  know  less  about  mathe- 
matics  than   the   "pure   mathematician" 


SCIENCE 


53 


does  about  practical  things,  and  they  ought 
to  cultivate  the  acquaintance  of  the  mathe- 
matician and  rub  off  the  worst  parts  of 
their  ignorance  before  they  attempt  to 
criticize  the  teaching  of  mathematics.  But 
it  is  much  easier  to  find  fault  and  say  that 
they  never  found  any  use  for  such  and 
such  mathematical  branches,  when  they 
never  gave  them  enough  attention  to  make 
them  of  any  use. 

Every  mathematical  teacher  should  teach 
all  three  parts  of  a  problem,  but  the  aver- 
age engineering  student  is  so  indifferent  to 
real  progress  and  his  limited  time  is  so 
taken  up  with  other  things  that  he  may  get 
through  his  course  knowing  very  little 
about  mathematics,  no  matter  how  well  it 
may  be  taught. 

Students  with  fair  ability  that  really 
want  to  learn  a  particular  subject  can  do 
it  even  under  indifferent  teachers,  but  un- 
less students  exert  themselves  to  learn,  the 
best  teacher  can  not  put  knowledge  into 
them.  Discuss  the  subject  to  the  limit, 
analyze  and  adjust  the  engineering  courses 
to  a  nicety,  write  new  text-books,  adopt 
new  systems  and  get  new  teachers  and  the 
thing  will  remain  about  as  it  is;  teachers 
will  teach  and  students  will  expect  them 
to,  while  only  a  few  will  learn,  whether 
the  teacher  expects  them  to  or  not. 

By  H.  T.  Eddy,  Dean  of  the  Graduate 
School   and   Professor   of   Mathematics 
and  Mechanics,  College  of  Engineering, 
University  of  Minnesota. 
Complaint  has  been  made  that  in  our 
teaching  of  mathematics  we  do  not  pay  due 
attention  to  psychological  and  pedagogical 
principles.     I  want  to  consider  for  a  mo- 
ment   the    application    of    two    of    these 
principles. 

First,  it  is  necessary  for  the  engineering 
student  to  have  an  ample  undergraduate 
course  in  mathematics,  and  such  an  ex- 
tended drill  in  and  habitual  acquaintance 


with  its  processes  that  when  he  has  for- 
gotten nine  tenths  of  it,  just  as  he  will  of 
this  and  all  other  subjects  which  he  studies 
in  college,  what  remains  with  him  will  be 
a  sufficient  equipment  in  this  line  for  his 
professional  career.  In  other  subjects  his 
residuum  of  knowledge  is  easily  refreshed 
and  increased.  Not  so  in  mathematics. 
The  stock  of  mathematical  knowledge  of 
which  he  is  easily  master  on  entering  his 
profession  will  practically  be  the  end  of 
his  attainments  in  that  direction.  Ke- 
stricting  the  course  in  mathematics  to  bare 
essentials  is  suicidal,  for  of  it  a  small  frac- 
tion only  will  remain  as  a  permanent  pos- 
session, and  that  fraction  is  likely  to  be 
smaller,  the  smaller  the  amount  originally 
attempted. 

Second,  the  teacher  of  mathematics  is 
prone  to  think  that  a  clear  presentation  of 
mathematical  truth  on  his  part,  and  a 
logical  demonstration  by  the  student,  are 
all  that  is  required  in  this  subject.  But 
important  as  these  things  assuredly  are, 
they  are  insufficient  to  produce  successful 
results.  The  question  is  one  in  which 
human  interest  is  really  of  more  im- 
portance than  logic,  for  mathematical 
knowledge  can  not  be  successfully  im- 
parted unless  genuine  interest  on  the  part 
of  the  student  can  be  in  some  way  aroused. 
It  goes  without  saying,  that  the  teacher 
must  first  of  all  have  that  interest  himself 
or  he  ceases  to  be  a  fit  teacher.  How  he 
will  awaken  interest  in  his  pupil  depends 
upon  his  own  personality.  Many  do  this 
by  help  of  problems  which  elucidate  and 
apply  the  principles.  Just  here  lies  the 
reason  for  the  usual  inability  of  pro- 
fessional engineers  to  teach  mathematics. 
They  have  no  interest  in  mathematics 
itself.  It  is  the  engineering  problem  alone 
that  interests  them.  To  this  matter  of 
interest,  or  the  lack  of  it,  may  be  traced 
the  failure  which  is  apt  to  attend  the  sepa- 
ration of  classes  into  divisions  according 


54 


SCIENCE 


to  scholarship,  for  in  that  case  the  divisions 
made  up  of  poor  students  lose  the  impetus 
to  be  derived  from  the  interest  which  the 
good  students  exhibit  in  their  work  in 
which  all  participate  to  some  degree. 

By  S.  M.  Barton,  Professor  of  Mathe- 
matics, University  of  the  South. 
While  standing  here  in  the  heart  of  the 
modern,  bustling  city  of  Chicago,  and 
listening  to  this  discussion,  my  mind  goes 
back  to  the  ancient  city  of  Tarentum  and 
her  distinguished  governor,  Archytas. 
Archytas,  while  an  able  mathematician, 
was  too  practical,  as  we  learn,  to  suit  the 
ideas  of  the  Platonic  School,  who  objected 
to  his  mechanical  solutions  of  certain 
mathematical  problems  as  interfering  with 
pure  reasoning.  Now,  while  I  take  an  im- 
mense interest  in  applied  mathematics 
(what  mathematician  at  this  day  would 
not?)  yet  I  confess  to  a  feeling  of  sym- 
pathy with  Plato  in  his  condemnation  of 
Archytas.  At  any  rate  I  wish  to  enter 
my  protest  against  a  possible  tendency  to 
degrade  mathematical  teaching  to  the 
memorizing  of  thumb-rules,  and  to  urge  the 
advantage  of  a  strong  backbone  of  pure 
mathematics  in  our  engineering  courses. 

I  read  with  interest  a  paper  presented 
at  the  Ithaca  meeting  of  the  Society  for 
the  Promotion  of  Engineering  Education, 
by  Professor  Arthur  E.  Haynes  of  the 
University  of  Minnesota,  in  justification  of 
the  use  of  the  expression  "engineering- 
mathematics."  I  must  say  I  was  at  first 
somewhat  shocked  by  the  expression,  for 
I  had  always  believed  that  mathematics  is 
mathematics  take  it  when  and  where  you 
will.  While  I  would  agree  heartily  with 
much  that  Professor  Haynes  said,  and  I  do 
not  doubt  that  his  courses  are  interesting 
and  instructive,  yet  I  question  the  wisdom 
of  drawing  any  sharp  distinction  in  the 
college    curriculum    between    the    mathe- 


matics given  to  the  engineering  student 
and  to  any  other  class  of  students. 

I  find  myself  differing  absolutely  from 
the  gentleman  from  the  Massachusetts  In- 
stitute of  Technology,  who  apparently  sees 
no  beauty,  much  less  utility,  in  the  higher 
branches  of  pure  mathematics.     How  Pro- 
fessor Woods,  who  has,  by  the  way,  written 
such  a  sound  text-book  on  mathematics, 
can  live  amicably  in  the  same  state,  much 
less  in  the  same  college,  as  his  engineer- 
colleague,  I  am  at  a  loss  to  understand- 
perhaps    they    have    an    occasional    fight. 
But,   joking  aside,   there  is  a  dangerous 
tendency  to  adopt  rules  (slide  and  mental) 
and  short-cut,  approximate  solution  to  the 
utter  exclusion  of  rigid  proofs.     Is  it  wise 
to  make  a  mere  machine  of  the  young  engi- 
neer,   even    if    thereby   he    becomes   rich 
faster  or  grows  poor  less  slowly  ?     I  freely 
admit,    however,    that    too    much    theory 
would  be  disastrous,  and  that  there  is  great 
room  for  improvement  in  the  teaching  of 
mathematics.      The    student    should    be 
taught  how  to  use  his  mathematics,  and  the 
existing  gap  between  theory  and  practise 
be  bridged.     While  affording  every  pos- 
sible facility  to  the  student  for  making  ex- 
periments, collecting  data,  becoming  expert 
in  handling  instruments,  making  calcula- 
tions, etc.,  I  urge  that  we  give  them,  one 
and  all,  a  good  rigid  course  in  pure  mathe- 
matics. 

By  Arthur  E.  Haynes,  Professor  of  En- 
gineering-Mathematics, University  of 
Minnesota. 

I  have  been  called  upon,  by  name,  to 
defend  the  use  of  the  term,  "Engineering 
Mathematics."  The  justification  of  the 
term  will  be  found  in  my  paper  on  the 
subject  in  Volume  XIV.  of  the  Proceed- 
ings of  the  Society  for  the  Promotion  of 
Engineering  Education.  As  the  paper 
was  not  read  before  this  association,  many 


SCIENCE 


55 


of  the  members  present  are  not  acquainted 
with  its  contents. 

In  brief,  the  reasons  there  given  for  the 
use  of  the  term  are: 

(a)  Because  of  the  main  object  of  the 
study  of  mathematics  in  engineering 
courses,  viz :  its  use  as  a  tool. 

(6)  Because  of  the  proper  method  of 
teaching  the  mathematics  of  such  courses. 

(c)  Because  of  the  content  of  the  mathe- 
matics of  such  courses. 

It  is  not  a  degradation  of  mathematics 
to  make  it  practical,  it  is  rather  an  added 
glory.  It  is  as  justifiable  to  use  this  term 
as  to  use  the  corresponding  terms  agricul- 
tural chemistry,  agricultural  botany,  engi- 
neering drawing,  etc.  We  do  not  degrade 
chemistry  or  botany  or  drawing  by  the 
use  of  these  terms:  but  their  employment 
is  justified  by  the  objects  of  the  study,  by 
the  methods  required  in  teaching  them  and 
by  their  content,  as  in  mathematics. 

It  has  been  suggested  that  a  less  thor- 
ough study  of  mathematics  is  advocated. 
In  reply  to  this,  may  I  quote  from  an 
article  in  Volume  VIII.  of  the  Proceed- 
ings of  the  Society  for  the  Promotion  of 
Engineering  Education,  on  "The  Teaching 
of  Mathematics  to  Engineering  Students," 
where  in  speaking  of  such  teaching  I  said : 

(a)  It  should  be  of  such  a  character  as 
to  produce  an  enduring  stimulating  effect 
upon  the  mind  of  the  student. 

(&)  It  should  give  the  student  the  power 
to  properly  interpret  mathematical  lan- 
guage, and  to  accurately  and  skillfully  use 
it. 

(c)  To  secure  these  results,  the  teaching 
must  be  based  upon  a  proper  order  of 
studies  and  carried  forward  in  a  rational, 
intelligent  manner. 

By   Arthur  S.  Hathaway,  Professor   of 

Mathematics,  Rose  Polytechnic  Institute. 

In  a  paper  on  "Pure  Mathematics  for 

Engineering  Students,"  published  in  the 


Bulletin  for  March,  1901,  I  expressed  opin- 
ions which  coincide  with  those  given  here 
to-day.  I  then  said  that  instruction  in 
mathematics  for  engineering  students 
should  have  two  objects  (1)  to  develop  an 
engineering  mind,  and  (2)  to  develop 
mathematics  as  an  instrument  of  research 
for  the  engineer.  I  came  to  these  conclu- 
sions at  that  time  as  a  result  of  inquiries 
made  of  graduates  of  several  institutions, 
who  were  in  engineering  practise,  and  of 
their  employers.  From  the  latter,  I  have 
had  the  statement  that  it  is  inadvisable  to 
place  a  man  in  the  higher  positions  in  engi- 
neering who  has  not  had  a  good  mathe- 
matical training,  especially,  in  the  cal- 
culus, which,  they  assert,  develops  those 
modes  of  thought  which  are  necessary  to 
the  engineer. 

I  wish  to  call  your  attention  to  the  fact 
that  the  fifty-four  hours  of  analytical 
dynamics  credited  to  Rose  Polytechnic  In- 
stitute on  this  chart  are  spent  on  applied 
calculus.  There  is  a  regular  course  of 
one  hundred  and  forty-four  hours  in 
Rankine  not  mentioned  here,  which  is 
given  by  my  colleague,  Professor  Gray. 
In  applied  calculus  we  take  up  problems 
which  require  the  use  of  the  calculus,  such 
as  motions  in  constant,  elastic  and  central 
fields,  the  bending  of  beams,  the  twisting 
of  shafts,  problems  in  electricity,  in  chem- 
istry, etc.  We  take  problems  gathered 
from  all  sources,  text-books,  magazines, 
engineering  professors,  and  discuss  them  in 
the  class-room,  with  special  reference  to 
the  analysis  and  its  mode  of  application. 

By  Edward  V.  Huntington,  Assistant 
Professor  of  Mathematics,  Harvard  Uni- 
versity. 

I  desire  to  call  attention  to  the  fact  that 
besides  the  analogy  of  mathematics  as  a 
tool  or  instrument,  there  is  also  the  perhaps 
more  significant  analogy  of  the  mathe- 
matician as  the  discoverer  of  quantitative 


56 


SCIENCE 


relations  which  already  exist  in  the  prob- 
lems themselves.  Logarithmic  relations 
between  varying  quantities,  for  instance, 
are  not  dragged  into  the  problem  from 
some  artificial  tool-chest,  but  are  already 
present  in  the  problem,  and  are  analyzed 
out  of  the  problem  much  as  the  precious 
metal  is  analyzed  out  of  the  ingot  by  the 
metallurgist.  The  practical  mathema- 
tician is  simply  a  scientist  specially  trained 
to  perceive  the  quantitative  aspects  of 
physical  phenomena. 

By  Donald  F.  Campbell,  Professor  of 
Mathematics,  Armour  Institute  of  Tech- 
nology. 

We  have  had  a  number  of  good  ideas 
set  before  us  in  the  last  two  days— ideas 
which  we  ought  to  make  an  effort  to 
crystallize.  I  think  that  the  present  time 
is  the  psychological  moment  to  have  a  com- 
mittee appointed  to  draw  up  a  report  on 
mathematics  for  colleges  of  engineering. 
This  report  perhaps  might  be  in  the  nature 
of  a  symposium,  but  it  would  be  especially 
valuable  if  it  considered  in  detail  the  sub- 
jects which  should  be  emphasized  in  a 
course  in  mathematics  for  engineering  stu- 
dents. These,  however,  are  merely  sug- 
gestions. I  would  not  hamper  the  com- 
mittee in  their  deliberations  by  outlining 
any  particular  course  which  they  should 
pursue.  The  only  condition  which  I  would 
impose  is  that  the  committee  be  representa- 
tive enough  that  all  of  us  can  look  towards 
their  report  with  the  utmost  confidence. 

I  would  move  that  the  chairman  be  em- 
powered to  appoint  a  committee  of  three, 
these  three  to  increase  their  number  to 
fifteen,  chosen  from  among  the  teachers  of 


mathematics  and  engineering  and  the 
practising  engineers,  these  fifteen  to  con- 
stitute a  committee  authorized  by  this 
meeting  to  make  such  a  report  on  mathe- 
matics for  colleges  of  engineering  as  in 
their  opinion  will  be  of  service  to  teachers 
in  such  institutions,  and  to  submit  this 
report  when  completed  to  the  Chicago  Sec- 
tion of  the  American  Mathematical  So- 
ciety.1 

1  Professor  Campbell's  motion,  as  amended  by 
Professor  Magruder,  requires  the  Committee  of 
Fifteen  to  report  to  the  Society  for  the  Promotion 
of  Engineering  Education  at  its  meeting  in  the 
summer  of  1909. 

The  chairman  appointed  Professors  Huntington, 
Williams  and  Townsend  as  the  committee  of  three. 
See  pages  2  and  3  of  this  report. 

The  committee  of  three  appointed  the  following 
persons  as  members  of  the  Committee  of  Fifteen: 
E.  V.  Huntington,  Harvard  University,  Cam- 
bridge, Mass.,  Chairman;  Philip  R.  Alger,  U.  S. 
Naval  Academy,  Annapolis,  Md.;  D.  F.  Campbell, 
Armour  Institute  of  Technology,  Chicago,  111.; 
E.  A.  Engler,  Worcester  Polytechnic  Institute, 
Worcester,  Mass.;  C.  N.  Haskins,  University  of 
Illinois,  Urbana,  111.;  C.  S.  Howe,  Case  School  of 
Applied  Science,  Cleveland,  Ohio;  Emil  Kuichling, 
New  York,  N.  Y.;  W.  T.  Magruder,  Ohio  State 
University,  Columbus,  Ohio;  Ralph  Modjeski, 
Chicago,  111.;  W.  F.  Osgood,  Harvard  University, 
Cambridge,  Mass.;  C.  S.  Slichter,  University  of 
Wisconsin,  Madison,  Wis.;  C.  P.  Steinmetz,  Sche- 
nectady, N.  Y. ;  G-.  F.  Swain,  Massachusetts  Insti- 
tute of  Technology,  Boston,  Mass.;  E.  J.  Town- 
send,  University  of  Illinois,  Urbana,  111.;  F.  E. 
Turneaure,  University  of  Wisconsin,  Madison, 
Wis.;  C.  A.  Waldo,  Washington  University,  St. 
Louis,  Mo.;  G.  S.  Williams,  University  of  Mich- 
igan, Ann  Arbor,  Mich.;  C.  M.  Woodward,  Wash- 
ington University,  St.  Louis,  Mo.;  R.  S.  Wood- 
ward, Carnegie  Institution,  Washington,  D.  C; 
Alexander  Ziwet,  University  of  Michigan,  Ann 
Arbor,  Mich. 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 
LOAN  DEPT. 

This  book  is  due  on  the  last  date  stamped  below,  or 

on  the  date  to  which  renewed. 

Renewed  books  are  subject  to  immediate  recall. 


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LD  21A-50m-8,'57 
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General  Library 

University  of  California 

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•.'»a82 


